Article

Robust Multiscale Identification of Apparent ElasticProperties at Mesoscale for Random HeterogeneousMaterials with Multiscale Field Measurements

Tianyu Zhang †, Florent Pled *,† and Christophe Desceliers *,†

Univ Gustave Eiffel, MSME UMR 8208, F-77454 Marne-la-Vallée, France; tianyu.zhang@univ-eiffel.fr* Correspondence: florent.pled@univ-eiffel.fr (F.P.); christophe.desceliers@univ-eiffel.fr (C.D.)† These authors contributed equally to this work.

Received: 16 April 2020; Accepted: 17 June 2020; Published: 23 June 2020

Abstract: The aim of this work is to efficiently and robustly solve the statistical inverse problem relatedto the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneousanisotropic materials with a complex microstructure that usually cannot be properly described in termsof their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparentelasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field.A general methodology using multiscale displacement field measurements simultaneously made at bothmacroscale and mesoscale has been recently proposed for the identification the hyperparameters of such aprior stochastic model by solving a multiscale statistical inverse problem using a stochastic computationalmodel and some information from displacement fields at both macroscale and mesoscale. This papercontributes to the improvement of the computational efficiency, accuracy and robustness of such amethod by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) ofkinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used ina previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representationof the hyperparameters involved in the prior stochastic model in order to enhance both the robustnessand the precision of the statistical inverse identification method. Finally, the proposed improved methodis first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity(using multiscale simulated data obtained through numerical computations) and then exemplified on areal heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linearelasticity (using multiscale experimental data obtained through mechanical testing monitored by digitalimage correlation).

Keywords: multiscale; mesoscale; statistical inverse problem; random heterogeneous materials; randomelasticity field; stochastic modeling

MSC: 62M40; 35J25; 60H15; 65C05; 65C20; 74B05; 74G75; 74S05; 74S60; 74Q05; 62P10; 62P30

1. Introduction

Within the framework of linear elasticity theory, the numerical modeling and simulation ofheterogeneous materials with hierarchical complex random microstructure give rise to many scientificchallenges. Their modeling is a topical issue with numerous applications in diverse material sciences,including for instance sedimentary rocks, natural composites, fiber- or nano-reinforced composites, someconcretes and cementitious materials, some porous media, some living biological tissues, among manyothers [1]. Although such materials are often considered and modeled as deterministic and homogeneous

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elastic media at macroscale in most practical applications, they are not only random and heterogeneous atmicroscale but they also usually cannot be explicitly described by any local morphological and mechanicalproperties of their constituents and easily reconstructed in a computational framework in the presenceof multiple interfaces. The modeling and identification of their elastic properties at meso- or microscaleshave been the subject of many research works in recent decades. Nowadays, with the recent developmentsachieved around the construction of stochastic models for tensor-valued random elasticity fields andtheir experimental inverse identification using field imaging techniques, one of the most promising waysconsists in introducing a prior stochastic model of the apparent elasticity tensor field of heterogeneousmaterials of the considered microstructure at a given mesoscale. Note that this mesoscopic scale allowsthe introduction of the spatial correlation length(s) of the microstructure, and that for materials with ahierarchical structure, such as cortical bone or tendon, different mesoscopic scales can be defined. Such amesoscopic stochastic modeling of random heterogeneous elastic media can further be used to characterizethe macroscopic mechanical properties in the context of the stochastic homogenization over a representativevolume element (RVE) subdomain. This representative volume element should be, provided that it exists,sufficiently large compared to the microscale and sufficiently small compared to the macroscale. In thepresent probabilistic context, a major question concerns the statistical inverse identification of a priorstochastic model parameterized by a small or moderate number of hyperparameters using only partialand limited experimental data.

1.1. Overview of Inverse Methods for the Mechanical Characterization of Micro/Meso-Structural Properties

The inverse methods for the experimental identification of elastic properties of homogeneous orheterogeneous materials at macroscale and/or mesoscale have been the subject of numerous researchworks over the three past decades. The first methods related to the experimental characterization anddescription of random microstructural morphologies by using image analysis techniques have beenintroduced and developed by the end of the 1980s [2–6] for the numerical modeling and simulationof random microstructures made up with heterogeneous materials. Since the early 1990s, significanttechnological advances in the field of optical measuring instruments, such as digital cameras equippedwith Charge-Coupled Device (CCD) or Complementary MetalOxideSemiconductor (CMOS) image sensorsand microscope objectives, have widely contributed to the emergence of imaging techniques such astwo-dimensional (2D) or three-dimensional (3D) digital image correlation (DIC) for identification purposes.DIC techniques [7–9] are now commonly used in solid mechanics and material sciences for experimentalmeasurements of elastic displacement fields of samples under external loading [10–16] in order to identifymechanical properties of complex microstructures for heterogeneous materials [13,17–24] with differentclasses of material symmetries. The recent milestones achieved around data acquisition systems andprocessing softwares for 3D images obtained for example by X-ray computed microtomography (µCT)[25–30], magnetic resonance imaging (MRI) [31–34], optical coherence tomography (OCT) [35–39] or anyother non-invasive and non-destructive testing technique for the reconstruction of 3D images in highresolution, have allowed the development of three-dimensional measurements of displacement fieldsby digital volume correlation (DVC) [9,15,40–50]. Such 3D full-field measurements offer the potentialof identifying stochastic models of 3D tensor-valued random elasticity fields at different scales for themechanical characterization of 3D real microstructures made up of heterogeneous materials.

In the mid 2000s, many research works have been carried out on the statistical inverse identificationof stochastic models of the tensor-valued random elasticity field in low or high stochastic dimension atmacroscopic and/or mesoscopic scale for complex microstructures modeled by random heterogeneousisotropic or anisotropic linear elastic media [51–66]. The proposed methodologies for solving the statisticalinverse problem related to the identification of a non-Gaussian tensor-valued random field in high

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stochastic dimension using available, partial and limited experimental data are mostly based on (i)the mathematical formulations of functional analysis for stochastic boundary value problems, (ii) thestatistical tools derived from probability theory, information theory, mathematical statistics and stochasticoptimization, such as the least-squares (LS) method [67,68], the maximum likelihood estimation (MLE)method [68–71], the maximum entropy (MaxEnt) principle [68,72–78], the nonparametric statistics [69,79],the Bayesian inference method [68,80–86], the statistical and computational inverse problems and relatedstochastic optimization algorithms [71,87–93], (iii) advanced functional representation techniques andprobabilistic methods, such as the Karhunen-Loève (KL) decomposition [94–96] to construct reduced-orderstochastic models, the polynomial chaos (PC) expansion [97–101] for an adapted high-dimensionalstochastic representation of non-Gaussian second-order random fields, (iv) the spectral methods[97,102–105] and sampling-based approaches [106–108] for solving stochastic boundary value problems,and (v) the stochastic homogenization methods [1,5,6,109–132] to bridge the meso- or microscopic scaleand the macroscopic scale. Combining such advanced probabilistic and statistical methods has led toearly fundamental works on the statistical inverse identification of non-Gaussian scalar- or tensor-valuedrandom fields in low or high stochastic dimension based on partial and limited experimental data. Theseworks have mainly been devoted to the statistical inverse identification of hyperparameters of priorstochastic models in low stochastic dimension, such as a mean field, a dispersion coefficient and somespatial correlation length(s) or the deterministic coefficients of a polynomial chaos expansion of therandom field [51–53,55–64,133–135]. To date, the latest and more advanced works focus on the inverseidentification of posterior stochastic models, that are high-dimensional stochastic representations of priorstochastic models for non-Gaussian scalar- or tensor-valued random fields [65,66,135–139].

1.2. Multiscale Statistical Identification Method

In keeping with the aforementioned works, an innovative methodology has been recently proposedin Reference [140] for the multiscale statistical inverse identification of a prior stochastic model of therandom apparent elasticity field at mesoscale for a heterogeneous anisotropic elastic microstructure. Thismultiscale identification procedure has been formulated within the framework of 3D linear elasticity theoryunder the following assumptions: (i) at macroscale, the elasticity tensor is deterministic and homogeneousand therefore independent of the spatial coordinates; (ii) at a given mesoscale, the tensor-valued randomelasticity field is the restriction to a mesoscopic subdomain of a statistically homogeneous random fieldindexed by R3, allowing to be consistent with the assumption for the existence of a representative volumeelement in the framework of stochastic homogenization [68,128].

The proposed method allows for the multiscale inverse identification of (i) the tensor-valued randomfield that models the apparent elasticity tensor field at a given mesoscale, and (ii) the effective elasticitytensor at macroscale, for a heterogeneous anisotropic elastic material with a random microstructurewhose morphological and mechanical properties cannot be properly described and reconstructed in acomputational framework from the local topology and mechanical behavior of its constitutive phases.The prior stochastic model of the random elasticity field is constructed by using the MaxEnt principle[68,72–78], initially derived within the general framework of information theory [141–143]. We then obtaina second-order mean-square continuous non-Gaussian positive-definite symmetric real matrix-valuedrandom field. In addition, an explicit algebraic representation has been established in Reference [144]. Sucha prior stochastic model of random elasticity field has been used, in particular, for stochastic boundaryvalue problems, such as static linear elasticity problems [68,128,144]. It is classically parameterized by asmall or moderate number of scalar-, vector- and/or tensor-valued hyperparameters, namely the meanfunction of the random elasticity field, a dispersion coefficient controlling the level of statistical fluctuationsof the random elasticity field around its mean function and spatial correlation lengths characterizing

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the spatial correlation structure of the random elasticity field. The statistical inverse problem for theidentification of this prior stochastic model is formulated as a multi-objective optimization problem forwhich the optimal parameters are the optimal values of the hyperparameters of the stochastic model.However, within the framework of this identification methodology, it can be shown that the mean functionof the random elasticity field cannot directly be identified using only the available experimental kinematicfield measurements at mesoscale. The experimental values of the stress fields associated with the kinematicfields observed experimentally at mesoscale should also be known, but these values are not availablein practice. Conversely, it can also be shown that the other hyperparameters (dispersion coefficient andspatial correlation lengths) controlling the statistical fluctuations of the random elasticity field cannotdirectly be identified using only the available experimental kinematic field measurements at macroscale.Consequently, such a statistical inverse identification procedure requires multiscale experimental fieldmeasurements that must be made simultaneously at both macroscopic and mesoscopic scales, since byassumption only a single specimen submitted to a given external loading at macroscale is experimentallytested. A stochastic homogenization method is then used to propagate the uncertainties at mesoscaletowards the macroscale under the classical assumption of scale separation between macroscale andmesoscale, so that a sufficiently large mesoscopic subdomain can be defined within the macroscopic domainand considered as a representative volume element. However, it should be noted that it is not necessaryfor this representative volume element to be the same size as the mesoscopic domain(s) of observation onwhich the experimental measurements are performed. Thus, the multiscale statistical inverse problemis formulated as a multi-objective optimization problem that consists in minimizing a (vector-valued)multi-objective cost function defined by three numerical indicators corresponding to single-objective costfunctions [140], namely (i) a macroscopic numerical indicator allowing the distance between the measuredexperimental fields and the computed numerical fields to be quantified at macroscale, (ii) a mesoscopicnumerical indicator allowing the distance between the statistical fluctuations exhibited by the measuredexperimental fields and the ones exhibited by the computed numerical fields to be quantified at mesoscale,and (iii) a multiscale numerical indicator allowing the distance between the elasticity tensor at macroscaleand the effective elasticity tensor constructed by computational stochastic homogenization of the randomapparent elasticity field in a representative volume element at mesoscale.

1.3. Drawbacks and Limitations of the Multiscale Identification Method

The multiscale identification method proposed in Reference [140] has been first validated by numericalsimulations on in silico materials and then successfully applied to the experimental characterization of theelastic properties of a biological tissue (beef cortical bone) within the framework of 2D plane stress linearelasticity from multiscale optical measurements of displacement fields performed at both macroscopicand mesoscopic scales on a single cortical bone specimen under static external loading at macroscale[145]. Nevertheless, the proposed identification method has some drawbacks that limit its use. First,it should be noted that the cost functions introduced for the multi-objective optimization problem arenot dedicated to a particular hyperparameter of the prior stochastic model of the random field to beidentified. Therefore, the only approach considered for solving the multi-objective optimization problemwas to use a global optimization algorithm (genetic algorithm) that belongs to the class of random search,genetic and evolutionary algorithms [146–156] to randomly explore the admissible set of hyperparameters.Despite a suitable parameterization (population size at each new generation, random generation of initialpopulation, selection procedure for reproduction including crossover and mutation operators, elite count,stopping criteria, etc.) of the genetic algorithm used in Reference [140] and the use of parallel processingand computing, the computational cost for solving the multi-objective optimization problem is high.This is due in particular to the large stochastic dimension of the tensor-valued random elasticity field.

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Secondly, during the validation and implementation of the multiscale identification method proposedin Reference [140], it was found that, for different mesoscopic domains of observation within the samemacroscopic domain, the resolution of the multi-objective optimization problem led to different optimalvalues of hyperparameters from one domain to another. Indeed, the experimental field measurementsover each mesoscopic domain of observation can be modeled as different random fields, and thereforethe multi-objective cost function on each mesoscopic domain of observation is a deterministic function ofthese random fields. This explains why the statistics of the multi-objective cost function are different fromone mesoscopic domain of observation to another. In Reference [140], the multi-objective cost function hasbeen replaced by the statistical average of the multi-objective cost functions calculated over each of themesoscopic domains of observation.

1.4. Improvements of the Multiscale Identification Method and Novelty of the Paper

In order to overcome the issues outlined above, this research work aims to present two majorimprovements of the methodology initially proposed in Reference [140] allowing the statistical inverseidentification of the tensor-valued random elasticity field at mesoscale to be performed with a bettercomputational efficiency, higher accuracy and improved robustness. First, we introduce an additionalmesoscopic numerical indicator allowing the distance between the spatial correlation length(s) of themeasured experimental kinematic fields and the one(s) of the computed numerical kinematic fields to bequantified at mesoscale, so that each hyperparameter of the prior stochastic model has its own dedicatedsingle-objective cost function, thus allowing the time-consuming global optimization algorithm (geneticalgorithm) used in Reference [140] to be avoided and replaced with a more efficient algorithm, suchas a fixed-point iterative algorithm, for solving the underlying multi-objective optimization problem.Secondly, in the case where experimental field measurements are available on several mesoscopic domainsof observation, we propose to not replace “naively” the multi-objective cost function by its empirical meanover all the mesoscopic domains of observation, but to consider a multi-objective optimization problem foreach mesoscopic domain of observation. Thus, each mesoscopic domain of observation leads to a possiblesolution of the values of the hyperparameters. Each of these values is then considered as a realization ofa random vector of hyperparameters whose prior stochastic model is constructed by using the MaxEntprinciple, and whose hyperparameters can be determined by using the MLE method, in order to improveboth the robustness and the accuracy of the inverse identification method of the prior stochastic model.

1.5. Outline of the Paper

The paper is organized as follows. Following this introduction, Section 2 presents the generalassumptions for solving the underlying multiscale statistical inverse problem. Then, Section 3 is dedicatedto the description of the multiscale experimental test configuration for obtaining experimental dataat both macroscale and mesoscale. Section 4 describes the prior stochastic model of the fourth-ordertensor-valued random elasticity field and its parameterization. Section 5 focuses on the objectives of themultiscale statistical inverse problem and the multiscale identification strategy. Next, Section 6 presents theconstruction of the macroscopic, mesoscopic and multiscale numerical indicators that are used for solvingthe multiscale statistical inverse problem as a multi-objective optimization problem. In this section, a focusis made on the improvements proposed by this paper in the definition of these numerical indicators withrespect to the previous work presented in Reference [140]. The multi-objective optimization problem isthen set in Section 7 and some numerical methods for solving such a multi-objective problem are presentedin Section 8. Section 9 discusses an improvement proposed in this paper for a robust identification whensome experimental field measurements are available on several mesoscopic domains of observation.Section 10 presents a numerical validation of the proposed multiscale identification methodology on in

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silico test specimens within the framework of 3D linear elasticity under 2D plane stress assumption and inthe general 3D case, for which the multiscale experimental data have been numerically simulated. Finally,Section 11 presents an experimental application to a real heterogeneous biological material constitutedof beef cortical bone within the framework of linear elasticity under 2D plane stress assumption, forwhich the multiscale experimental data have been obtained from a single static uniaxial compression testperformed on a specimen of beef femoral cortical bone and monitored by 2D digital image correlation atboth macroscale and mesoscale. Lastly, Section 12 gives some conclusions and potential perspectives ofthis work.

2. Assumptions for Solving the Multiscale Statistical Inverse Problem

In the present work, we address the statistical inverse identification of the elastic properties for acomplex microstructure made up of a heterogeneous anisotropic material and considered as a randomlinear elastic medium. In this section, we first state suitable assumptions for solving this multiscalestatistical inverse problem. Within the framework of linear elasticity theory, probability theory andcomputational stochastic homogenization in micromechanics and multiscale mechanics of heterogeneousmaterials, the following assumptions related to scale separation, stationarity and ergodicity properties areintroduced:

• there exists a scale separation between macroscale and mesoscale, so that a mesoscopic subdomaincan be defined and for which the dimensions are sufficiently large with respect to the size of theheterogeneities and sufficiently small with respect to the size of the macroscopic domain. Such amesoscopic subdomain can then be considered as a representative volume element;

• the random apparent elasticity tensor field at mesoscale is the restriction to one or more boundedmesoscopic subdomain(s) of a second-order stationary random field indexed by R3, and consequentlythe mean function of the random elasticity field at mesoscale is independent of the spatial coordinates;

• the random apparent elasticity tensor field at mesoscale is ergodic in average in the mean-squaresense, so that the homogenized elasticity tensor at macroscale calculated by stochastic homogenizationof the random apparent elasticity field in a mesoscopic subdomain corresponding to a representativevolume element can be considered as almost deterministic, in the sense that (i) its spatial averagereaches an asymptotic convergence with a very high level of probability for a sufficiently largemesoscopic subdomain size, and therefore (ii) its level of statistical fluctuations around its meanfunction at macroscale can be considered as negligible, thus yielding a deterministic homogenizedelasticity tensor at macroscale.

In this work, we focus on the class of heterogeneous materials that can be considered as random elasticmedia and for which the hypothesis stated on the scale separation between macroscale and mesoscaleis verified. It should be noted that, if such a scale separation assumption was not satisfied, then themultiscale statistical inverse problem under consideration would be an ill-posed problem if only a singleexperimental field measurement at macroscale was available, because in this case the macroscopic elasticity(or compliance) tensor must be modeled by a random tensor and a single experimental measurementis not sufficient to identify its stochastic model. The proposed identification methodology is thereforenot adapted to this case and would require several experimental field measurements at macroscale aswell as modifications of the macroscopic and multiscale indicators introduced in Section 6, and also theintroduction of additional numerical indicators at macroscale. Hereinafter, since the present identificationmethodology is developed within the framework of linear elasticity theory, we will use the terminology“strain field” to make reference to the “linearized strain field” for the sake of conciseness.

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3. Multiscale Experimental Test Configuration

The difficulties related to the acquisition of the experimental measurements for the inverseidentification procedure to be carried out are induced not only by the complex nature of the heterogeneousanisotropic elastic microstructure but also by the need to obtain multiscale kinematic field measurementsat two different scales (macroscale and mesoscale) for a single test specimen under given static loadingconditions through a multiscale DIC performed simultaneously at both macroscale and mesoscale. Toovercome such difficulties, a suitable experimental protocol, including the preparation of the test specimen,the development of a measuring bench, the acquisition system of digital images and the DIC method, hasbeen set up in Reference [145] for the acquisition of 2D multiscale optical measurements of displacementfields performed at both macroscale and mesoscale on a single beef cortical bone specimen submittedto a static vertical uniaxial compression test. Such a living biological tissue with a complex hierarchicalmicrostructure is of particular interest in the present context of multiscale modeling and identificationfor random heterogeneous materials. The multiscale experimental test configuration is briefly recalledhere. A sketch of the multiscale experimental configuration of the specimen at macroscale and mesoscaleis represented in Figure 1.

f macro

Ωmacroexp

ΓmacroN

ΓmacroD

∂Ωmacroexp

umacroexp

Ωmesoexp,q

∂Ωmesoexp,q

umesoexp,q

Figure 1. Multiscale experimental configuration: displacement field umacroexp measured in the macroscopic

domain of observation Ωmacroexp and displacement field umeso

exp,q measured in each mesoscopic domain ofobservation Ωmeso

exp,q, for q = 1, . . . , Q.

The test specimen has a cubic shape and is submitted to a simple external load. On the upper sideof the specimen, a surface force field is applied, while the opposite side of the specimen is clamped.Then, during the same and unique experimental loading, the displacement fields at both macroscale andmesoscale are simultaneously measured, for instance in using two optical digital cameras equipped withCCD imaging sensors with different spatial resolutions for the simultaneous acquisition of displacementfield optical measurements at both macroscopic and mesoscopic scales. The measurements are performedon the domain Ωmacro

exp at macroscale and on the domain Ωmesoexp at mesoscale that are 2D or 3D parts

of the specimen at macroscale and mesoscale, respectively. These domains can be 3D in the case ofmicrotomography techniques for the acquisition of 3D experimental data, or they can be 2D in thecase of digital camera techniques for the acquisition of 2D experimental data. Note that in case the

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dimensions of the mesoscopic domain of observation Ωmesoexp are very small with respect to the dimensions

of the macroscopic domain of observation Ωmacroexp , then more information can be used by collecting

additional experimental field measurements at mesoscale on Q non-overlapping mesoscopic domainsof observation Ωmeso

exp,1, . . . , Ωmesoexp,Q for which the relative mutual locations into the test specimen are not

necessarily recorded. The experimental database is then constituted of the vector-valued experimentaldisplacement fields umacro

exp and umesoexp,1, . . . , umeso

exp,Q, respectively, at macroscale on Ωmacroexp and at mesoscale on

Ωmesoexp,1, . . . , Ωmeso

exp,Q. The experimental tensor-valued strain fields εmacroexp and εmeso

exp,1, . . . , εmesoexp,Q, respectively

associated to the experimental displacement fields umacroexp and umeso

exp,1, . . . , umesoexp,Q, can be calculated by

post-processing through interpolation techniques.

4. Prior Multiscale Stochastic Model and Its Hyperparameters

At the macroscale, the specimen under test is modeled as a deterministic homogeneous linear elasticmedium for which the effective mechanical properties are represented by a deterministic model of thefourth-order elasticity tensor Cmacro(a) that is independent of spatial position x and parameterized bya vector a belonging to an admissible set Amacro. The vector-valued parameter a is constituted of thealgebraically independent coefficients spanning the macroscopic elasticity tensor Cmacro(a) having a givensymmetry class induced by linear elastic material symmetries. At the mesoscale, the specimen under test ismodeled as a random heterogeneous linear elastic medium for which the apparent mechanical propertiesare represented by a prior stochastic model of the fourth-order tensor-valued random elasticity field. InReference [144], the ensemble SFE+ of non-Gaussian second-order stationary random fields has beenintroduced and constructed in using the theory of information, the MaxEnt principle and the theory ofrandom matrices. Such a family of tensor-valued random fields is completely parameterized by the valuesof their mean function, a dispersion coefficient usually denoted as δ, and d n(n + 1)/2 = (d3(d + 1)2 +

2 d2(d + 1))/8 = 63 possibly different spatial correlation lengths, with d = 3 and n = d(d + 1)/2 = 6 in3D linear elasticity (see References [128,144] for a definition of the spatial correlation lengths of a randomfield). All these parameters are independent of the spatial position x since every tensor-valued randomfield in SFE+ is second-order stationary on R3 by construction. In addition, the dispersion coefficient δ

introduced in Reference [144] is such that

0 6 δ < δsup, with δsup =√(n + 1)/(n + 5) =

√7/11 ≈ 0.7977 < 1, (1)

where n = d(d + 1)/2 = 6 with d = 3 in 3D linear elasticity. Hence, any tensor-valued random fieldin SFE+ has no statistical fluctuations when δ = 0 and consequently its values are almost surely (a.s.)equal to its mean function. In addition, the level of statistical fluctuations of any tensor-valued randomfield in SFE+ increases with the value of δ. Consequently, the highest statistical fluctuations are obtainedwhen δ = δsup. Ensemble SFE+ has been especially constructed in Reference [144] for offering a priorstochastic model that can be used for modeling the tensor-valued apparent elasticity (or compliance) fieldsat mesoscale. Consequently, in this paper, we will use the same approach and the prior stochastic model ofthe elasticity tensor field Cmeso (resp. the compliance tensor field Smeso) will be defined as the restrictionto a given bounded subdomain in R3 of a random tensor field belonging to SFE+ and indexed by R3.The prior stochastic model of Cmeso or Smeso can then be deduced from each other by inverse of eachother. In this work, we will only consider the special case for which the spatial correlation structure ofCmeso (resp. Smeso) is defined by only 3 (instead of 63) different values `1, `2, `3 for the spatial correlationlengths and consequently some of the 63 spatial correlation lengths are mutually equal to each other.Furthermore, the mean function of Cmeso (resp. Smeso) can be represented by a set of nsym 6 n(n + 1)/2parameters h1, . . . , hnsym that might have or not physical meaning in mechanical engineering such asYoung’s moduli, Poisson’s ratios, bulk and shear moduli, and so forth (see for instance Section 10). Finally,

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the hyperparameters of the prior stochastic model of Cmeso (resp. Smeso) are δ, `1, `2, `3 and h1, . . . , hnsym

that can be gathered into the vector-valued hyperparameter b = (δ, `, h) in which ` = (`1, `2, `3) andh = (h1, . . . , hnsym). Hereinafter, the set of all the admissible values of vector h is denoted by Hmeso andthe admissible set of vector b is denoted by Bmeso.

5. Objectives and Strategy for Solving the Multiscale Statistical Inverse Problem

5.1. Objectives of the Multiscale Statistical Inverse Problem

The deterministic model of Cmacro(a) at macroscale and the prior stochastic model of Cmeso(b) atmesoscale have to be identified by calculating the optimal values amacro and bmeso of the vector-valuedparameter a ∈ Amacro and the vector-valued hyperparameter b ∈ Bmeso, respectively, according tothe experimental kinematic field measurements available at both macroscale and mesoscale. While thevector-valued parameter a can completely be identified by solving a usual deterministic inverse problemusing only the available experimental field measurements at macroscale, the vector-valued hyperparameterb = (δ, `, h) cannot directly be identified by solving a statistical inverse problem using only the availableexperimental field measurements at mesoscale. More precisely, the dispersion parameter δ and the vectorof spatial correlation lengths ` require only experimental field measurements at mesoscale to be identified,whereas the vector h requires additional experimental field measurements at macroscale to be identified.Indeed, the hyperparameters δ and ` controlling respectively the level of statistical fluctuations andthe spatial correlation structure of the random elasticity field require experimental field measurementswith a sufficiently fine spatial resolution to be identified, while the hyperparameters h representingthe mean elasticity field would require the experimental values of the stress fields associated with thekinematic (displacement or strain) fields observed experimentally at mesoscale to be identified, but thesevalues are not available in practice. The complete statistical information on random field Cmeso(b) mustthen be transferred to the macroscale in order to identify its mean function Cmeso using the availableexperimental field measurements at macroscale. A natural choice for such a transfer of informationconsists in computing the effective elasticity tensor Ceff(b) by a computational stochastic homogenizationmethod and in comparing it with the previously identified elasticity tensor Cmacro(a). Thus, unlike thevector-valued parameter a, the vector-valued hyperparameter b requires multiscale experimental fieldmeasurements (at macroscale and mesoscale) to be completely identified, thus leading to a challengingmultiscale statistical inverse problem to be solved. Since by assumption only a single specimen isexperimentally tested under a given static external loading applied at macroscale, the experimentalfield measurements must be performed simultaneously at both macroscale and mesoscale on the singletest specimen, but they do not need to be performed on the whole domain of the specimen.

5.2. Strategy for Solving the Multiscale Statistical Inverse Problem

Due to the major difficulties stated above and induced by the complexity of the challenging multiscalestatistical inverse problem to be solved, a first complete methodology concerning such a multiscaleidentification has been recently proposed in Reference [140], in which a multiscale statistical inverseidentification strategy is introduced and developed for an elastic microstructure with heterogeneousanisotropic statistical fluctuations within the framework of 3D linear elasticity theory. The proposedstrategy allows for the identification of (i) the optimal value amacro of vector-valued parameter a, and (ii)the optimal value bmeso of vector-valued hyperparameter b, by using the experimental displacement fieldmeasurements at both macroscale and mesoscale. The multiscale experimental identification methodologyoriginally developed in Reference [140] consists in introducing and constructing three different numericalindicators allowing the multiscale statistical inverse problem to be formulated as a multi-objective

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optimization problem. In the present work, we develop an improved multiscale experimental identificationmethodology involving four numerical indicators that are sensitive to the variation of the parameters andhyperparameters to be identified, which are:

1. A macroscopic numerical indicator J macro(a), dedicated to the identification of parameter a,that allows for quantifying the distance between the experimental strain field εmacro

exp associatedto the experimental displacement field umacro

exp measured at macroscale in the macroscopic domainΩmacro

exp and the strain field εmacro(a) associated to the displacement field umacro(a) computed froma deterministic homogeneous linear elasticity boundary value problem (with both Dirichlet andNeumann boundary conditions) that models the experimental test configuration at macroscale andinvolves the unknown deterministic elasticity tensor Cmacro(a);

2. A mesoscopic numerical indicator J mesoδ (b), dedicated to the identification of hyperparameter δ,

that allows for quantifying the distance between a pseudo-dispersion coefficient δεexp modeling the

level of spatial fluctuations of the experimental strain field εmesoexp associated to the experimental

displacement field umesoexp measured at mesoscale in a mesoscopic domain of observation Ωmeso

exp , and arandom pseudo-dispersion coefficient DE (b) representing the level of statistical fluctuations of therandom strain field Emeso(b) associated to the random displacement field Umeso(b) computed froma stochastic heterogeneous linear elasticity boundary value problem (with only Dirichlet boundaryconditions) that models the experimental test configuration at mesoscale and involves the randomelasticity tensor field Cmeso(b) with an unknown level of statistical fluctuations δ that must beidentified;

3. Another mesoscopic numerical indicator J meso` (b), dedicated to the identification of hyperparameter

` = (`1, `2, `3), that allows for quantifying the distance between the 3 different pseudo-spatialcorrelation lengths `ε

exp,1, `εexp,2, `ε

exp,3 of the experimental strain field εmesoexp in each spatial direction,

measured at mesoscale in a mesoscopic domain of observation Ωmesoexp , and the 3 pseudo-spatial

correlation lengths LE1 (b), LE

2 (b), LE3 (b) of the random strain field Emeso(b) in each spatial direction,

computed from the same mesoscopic stochastic boundary value problem as forJ mesoδ (b) for which the

random elasticity tensor field Cmeso(b) has a spatial correlation structure induced and characterizedby an unknown vector of spatial correlation lengths ` = (`1, `2, `3) that must be identified;

4. A multiscale numerical indicator J multih (a, b), dedicated to the identification of hyperparameter h,

that allows for quantifying the distance between the homogeneous deterministic elasticity tensorCmacro(a) at macroscale and the effective elasticity tensor Ceff(b) resulting from a computationalstochastic homogenization in a representative volume element ΩRVE at mesoscale of the randomelasticity tensor field Cmeso(b) whose mean function Cmeso is unknown and must be identify.

The multiscale statistical inverse problem then consists in identifying the optimal values amacro

and bmeso of the parameters a in Amacro and hyperparameters b in Bmeso, respectively, by solving amulti-objective optimization problem that consists in minimizing the (vector-valued) multi-objective costfunction J (a, b) =

(J macro(a),J meso

δ (b),J meso` (b),J multi

h (a, b))

involving the four aforementionednumerical indicators. However, for further computational savings, the multi-objective optimizationproblem can be decomposed into (i) a single-objective optimization problem that consists in minimizingJ macro(a) for identifying the optimal vector-valued parameter amacro using only the experimentalfield measurements at macroscale, and (ii) a multi-objective optimization problem that consistsin minimizing J meso(b) =

(J meso

δ (b),J meso` (b),J multi

h (amacro, b))

for identifying the optimalvector-valued hyperparameter bmeso using the experimental field measurements at mesoscale andexploiting the optimal vector-valued parameter amacro previously identified at step (i).

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6. Construction of the Numerical Indicators for Solving the Multiscale Statistical Inverse Problem

In this section, the construction of the macroscopic, mesoscopic and multiscale numerical indicatorsfor solving the multiscale statistical inverse problem is presented.

6.1. Deterministic Macroscopic Boundary Value Problem for the Macroscopic Indicator

At macroscale, the deterministic boundary value problem modeling the experimental testconfiguration described in Section 3 is written over an open bounded domain Ωmacro ⊂ R3 withmacroscopic dimensions of the specimen. The experimental domain of observation Ωmacro

exp is simulated asone given 2D or 3D part Ωmacro

obs of Ωmacro. The boundary ∂Ωmacro of Ωmacro consists of two disjoint andcomplementary parts Γmacro

N , on which Neumann boundary conditions are applied, and ΓmacroD , on which

Dirichlet boundary conditions are applied, such that ∂Ωmacro = ΓmacroN ∪ Γmacro

D and ΓmacroN ∩ Γmacro

D = ∅,with |Γmacro

D | 6= 0, where |ΓmacroD | denotes the 2D measure of Γmacro

D . A given deterministic surface forcefield f macro is applied on Γmacro

N , while homogeneous Dirichlet conditions are applied on ΓmacroD , so that

there is no rigid body motion during the test. Within the context of linear elasticity theory, the deterministicboundary value problem at macroscale consists in finding the vector-valued displacement field umacro andthe associated tensor-valued Cauchy stress field σmacro satisfying the following equilibrium equations,stress-strain constitutive equation and Neumann and Dirichlet boundary conditions

−div(σmacro) = 0 in Ωmacro, (2)

σmacro = Cmacro(a) : εmacro in Ωmacro, (3)

σmacro · nmacro = f macro on ΓmacroN , (4)

umacro = 0 on ΓmacroD , (5)

in which div denotes the divergence operator of a second-order tensor-valued field with respect to x,the colon symbol : denotes the classical twice contracted tensor product, nmacro is the unit normal vectorto ∂Ωmacro pointing outward Ωmacro and εmacro is the classical tensor-valued strain field associated todisplacement field umacro and defined by

εmacro = ε(umacro) =12

(∇ umacro + (∇ umacro)T

), (6)

in which ε denotes the deterministic linear operator mapping the displacement field to the correspondinglinearized strain field, the superscript T denotes the transpose operator and ∇ denotes the gradientoperator of a vector-valued field with respect to x. Recall that, as the material is assumed to be deterministicand homogeneous at macroscale, the unknown fourth-order deterministic elasticity tensor Cmacro(a)involved in constitutive Equation (3) is independent of x and parameterized by a parameter a belongingto an admissible set Amacro depending on the considered material symmetry class. A sketch of thedeterministic boundary value problem at macroscale is represented in Figure 2a.

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f macro

Ωmacro

ΓmacroN

ΓmacroD

∂Ωmacro

Cmacro(a)

umacro(a)

(a)

f macro

Ωmacro

ΓmacroN

ΓmacroD

∂Ωmacro

Cmacro(a)

umacro(a)

umesoexp

Ωmeso

∂Ωmeso

Cmeso(b)

Umeso(b)

(b)Figure 2. Boundary value problems at (a) macroscale and (b) mesoscale. (a) Deterministic boundaryvalue problem characterized by deterministic elasticity tensor Cmacro(a) at macroscale: deterministicdisplacement field umacro(a) computed at macroscale in Ωmacro; (b) Stochastic boundary value problemcharacterized by random elasticity tensor field Cmeso(b) at mesoscale: random displacement field Umeso(b)computed at mesoscale in Ωmeso.

6.2. Stochastic Mesoscopic Boundary Value Problem for the Mesoscopic Indicators

At mesoscale, the stochastic boundary value problem modeling the experimental test configurationdescribed in Section 3 is written over an open bounded domain Ωmeso ⊂ R3 with mesoscopic dimensions.A given domain of observation Ωmeso

exp corresponds to one given 2D or 3D part Ωmesoobs of Ωmeso. Within the

context of linear elasticity theory, the stochastic boundary value problem at mesoscale consists in findingthe vector-valued random displacement field Umeso and the associated tensor-valued random Cauchystress field Σmeso satisfying the following equilibrium equations, stress-strain constitutive equation andDirichlet boundary conditions

−div(Σmeso) = 0 in Ωmeso, (7)

Σmeso = Cmeso(b) : Emeso in Ωmeso, (8)

Umeso = umesoexp on ∂Ωmeso, (9)

where Emeso is the tensor-valued random strain field associated to random displacement field Umeso anddefined by

Emeso = ε(Umeso) =12

(∇Umeso + (∇Umeso)T

). (10)

Note that non-homogeneous Dirichlet boundary conditions (9) are prescribed on the whole boundary∂Ωmeso of Ωmeso, which correspond to the displacement field umeso

exp that is experimentally measured over agiven domain of observation Ωmeso

exp on the test specimen at mesoscale. Note also that (8) can equivalentlybe rewritten as

Σmeso = (Smeso(b))−1 : Emeso in Ωmeso, (11)

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where Smeso(b) = (Cmeso(b))−1 is the random compliance tensor field of the considered materialat mesoscale. For some linear elasticity problems, such as with 2D plane stress assumption,constitutive Equation (11) is more appropriate than (8). A sketch of the stochastic boundary valueproblem at mesoscale is represented in Figure 2b.

6.3. Macroscopic Numerical Indicator

Within the context of inverse identification, the optimal identified value amacro of parameter a can bedetermined by exploiting the sensitivity of the model strain field εmacro with respect to a and using theexperimental strain field εmacro

exp , which is obtained in Ωmacroexp but can be rewritten in Ωmacro

obs , through theintroduction of a macroscopic numerical indicator J macro(a) defined for any vector a ∈ Amacro by

J macro(a) =1

|Ωmacroobs |

∫Ωmacro

obs

‖εmacro(x; a)− εmacroexp (x)‖2

F dx, (12)

where |Ωmacroobs | denotes the measure of domain Ωmacro

obs and ‖ · ‖F denotes the Frobenius (or Hilbert-Schmidt)norm. Macroscopic numerical indicator J macro(a) allows for quantifying the spatial average over themacroscopic domain Ωmacro

obs of the distance between the model strain field εmacro(a) and the experimentalstrain field εmacro

exp at macroscale. The optimal vector-valued parameter amacro can then be identified byminimizing J macro(a) over all vector-valued parameter a in Amacro, provided that the model strain fieldεmacro(a) computed by solving the deterministic boundary value problem (2)-(6) is sufficiently sensitive toparameter a.

6.4. Mesoscopic and Multiscale Numerical Indicators

Within the context of statistical inverse identification, the optimal identified values bmeso =

(δmeso, `meso, hmeso) of b = (δ, `, h) can be determined by exploiting the sensitivity of some quantitiesof interest of the stochastic boundary value problem (7)-(10) with respect to δ, ` = (`1, `2, `3) and h,respectively, and using their counterparts coming from the experimental measurements through theintroduction of two mesoscopic numerical indicators J meso

δ (b) and J meso` (b) and one multiscale numerical

indicator J multih (a, b).

6.4.1. Mesoscopic Numerical Indicator Associated to the Dispersion Parameter

A first mesoscopic numerical indicator J mesoδ (b) is introduced to identify the dispersion parameter δ

controlling the level of statistical fluctuations of random elasticity field Cmeso(b) at mesoscale and definedfor any vector b ∈ Bmeso by

J mesoδ (b) =

(EDE (b) − δε

exp

δεexp

)2

, (13)

where E denotes the mathematical expectation, DE (b) is a positive-valued random variable that modelsthe random level of spatial fluctuations of the random solution obtained by solving the stochastic boundaryvalue problem (7)-(10) at mesoscale and where δε

exp is its counterpart for the experimental test specimen atmesoscale, such that

DE (b) =

√VE (b)

‖Emeso(b)‖Fand δε

exp =

√Vε

exp

‖εmesoexp ‖F

, (14)

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where Emeso(b) and εmesoexp are the spatial averages of random strain field Emeso(b) and experimental strain

field εmesoexp , respectively, and where

VE (b) =1

|Ωmesoobs |

∫Ωmeso

obs

‖Emeso(x; b)− Emeso(b)‖2F dx, (15)

Vεexp =

1|Ωmeso

obs |

∫Ωmeso

obs

‖εmesoexp (x)− εmeso

exp ‖2F dx, (16)

where |Ωmesoobs | denotes the measure of domain Ωmeso

obs . Note that it can easily be shown that Emeso(b) =εmeso

exp for all b ∈ Bmeso a.s. and consequently Emeso(b) is a deterministic tensor. Also, since randomstrain field Emeso(b) is a priori nor statistically homogeneous neither ergodic in average, Emeso(b) doesnot correspond to the statistical mean function of Emeso(b) and therefore VE (b) (resp. DE (b)) does notcorrespond to the variance (resp. dispersion coefficient) of Emeso(b). The mesoscopic numerical indicatorJ meso

δ (b) defined by (13) allows for quantifying the relative distance between the statistical mean value ofDE (b) and its experimental observation δε

exp. It should also be noted that a mesoscopic numerical indicatorsimilar to this one was introduced in Reference [140], but with different expressions than that of (13), (15)and (16) for the definitions of J meso

δ (b) and VE (b), respectively.

6.4.2. Mesoscopic Numerical Indicator Associated to the Spatial Correlation Lengths

A second mesoscopic numerical indicator J meso` (b) is introduced to identify the vector of spatial

correlation lengths ` = (`1, `2, `3) characterizing the spatial correlation structure of random elasticity fieldCmeso(b) (or random compliance field Smeso(b)) and defined for any vector b ∈ Bmeso by

J meso` (b) =

3

∑α=1

(ELE

α (b) − `εexp,α

`εexp,α

)2

, (17)

where LEα (b) is a positive-valued random variable that models the spatial correlation length along

the α-th spatial direction (relative to the spatial coordinate xα) characterizing the spatial correlationstructure of the statistical fluctuations of random strain field Emeso(b) and where `ε

exp,α is its observationfor the experimental test specimen at mesoscale. Usual signal processing methods (such as theperiodogram method) are used for estimating LE

α (b) and `εexp,α by considering the approximation

that they are independent of x which is not the case since Emeso(b) and εmesoexp are usually not

statistically homogeneous because of the non-homogeneous Dirichlet boundary conditions (9) involvingthe experimental displacement field umeso

exp on ∂Ωmeso. The mesoscopic numerical indicator J meso` (b)

defined by (17) allows for quantifying the relative distance between the statistical mean values ofLE

1 (b), LE2 (b), LE

3 (b) and their experimental observations `εexp,1, `ε

exp,2, `εexp,3.

6.4.3. Multiscale Numerical Indicator Associated to Computational Stochastic Homogenization

A multiscale numerical indicator J multih (a, b) is introduced to identify the mean function Cmeso(b) of

the random elasticity field Cmeso(b) at mesoscale and defined for any vector a ∈ Amacro and any vectorb ∈ Bmeso by

J multih (a, b) =

(‖ECeff(b) − Cmacro(a)‖F

‖Cmacro(a)‖F

)2

, (18)

where Ceff(b) is the effective elasticity tensor constructed by computational stochastic homogenization ofCmeso(b) in an open bounded mesoscopic domain ΩRVE, which is assumed to be a representative volume

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element. It should be noted that, under scale separation assumption, Ceff(b) is actually a random tensorfor which the level of statistical fluctuations tends to zero when the size of domain ΩRVE tends to infinity[68,128,131]. This is the reason why the statistical mean value ECeff(b) has been considered in thedefinition (18) of J multi

h (a, b) instead of the effective elasticity tensor Ceff(b) itself. The multiscale indicatorJ multi

h (a, b) defined by (18) allows for quantifying the relative distance between (i) the macroscopicelasticity tensor Cmacro(a) involved in the deterministic boundary value problem (2)-(6) at macroscale,and (ii) the statistical mean value of the effective elasticity tensor Ceff(b) calculated by a computationalstochastic homogenization method in the mesoscopic subdomain ΩRVE of the random elasticity fieldCmeso(b) involved in the stochastic boundary value problem (7)-(10) at mesoscale.

6.5. Comments

It should be noted that in the original formulation initially proposed [140], the numerical indicatorJ meso` (b) was not introduced. The improved formulation proposed in the present work is more advanced

than the original formulation initially proposed in Reference [140] to the extent that it involves an additionalmesoscopic numerical indicator, namely J meso

` (b), so that the parameter a and the three components δ,` and h of the hyperparameter b each have their own dedicated numerical indicator. Thus, the numberof single-objective cost functions being equal to the number of parameters to optimize, it is possible tosubstitute the computationally expensive global search algorithm used in Reference [140], which belongsto the class of random search, genetic and evolutionary algorithms [146–156], with a more computationallyefficient optimization algorithm, such as the fixed-point iterative algorithm considered in the presentwork (see Section 8). Indeed, even using parallel processing and computing tools, the computationalcost incurred by the global optimization algorithm (genetic algorithm) used in Reference [140] remainshigh due to the large stochastic dimension of the tensor-valued random elasticity field Cmeso(b), sothat the multi-objective optimization problem can be numerically intractable, with the current availablecomputer resources, in very high stochastic dimension for large-scale (non-)linear computational models ofthree-dimensional random microstructures. The computational cost of the genetic algorithm is comparedto the one of the fixed-point iterative algorithm in terms of the number of evaluations of the stochasticcomputational model in the 2D validation example presented in Section 10.1. It provides a measure of thecomputational efficiency that is independent of the computer hardware used to perform the numericalsimulations. Lastly, it should be noted that an alternative mesoscopic numerical indicator J meso

δ (b) isused compared to the previous work in Reference [140] without degrading the performance in terms ofaccuracy.

7. Multiscale Statistical Inverse Problem Formulated as a Multi-Objective Optimization Problem

The multiscale statistical inverse identification of parameter a and hyperparameter b can be performedsimultaneously by formulating the multiscale statistical inverse problem as a multi-objective optimizationproblem, that is

(amacro, bmeso) = arg mina∈Amacro,b∈Bmeso

J (a, b), (19)

where J (a, b) is the (vector-valued) multi-objective cost function consisting of the four aforementionednumerical indicators as single-objective cost functions and defined for any vector a ∈ Amacro and anyvector b ∈ Bmeso by

J (a, b) =(J macro(a),J meso

δ (b),J meso` (b),J multi

h (a, b))

. (20)

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In accordance with the strategy for solving the multiscale statistical inverse problem (see Section 5.2),for a better computational efficiency, the multiscale statistical inverse identification of a and b is performedsequentially by splitting the multi-objective optimization problem into two subproblems solved one afterthe other:

1. a macroscale inverse problem formulated as a single-objective optimization problem that consistsin calculating the optimal value amacro of parameter a in Amacro that minimizes the macroscopicnumerical indicator J macro(a), that is

amacro = arg mina∈Amacro

J macro(a); (21)

2. a mesoscale statistical inverse problem formulated as a multi-objective optimization problem thatconsists in calculating the optimal value bmeso of hyperparameter b in Bmeso that minimizes thetwo mesoscopic numerical indicators J meso

δ (b) and J meso` (b) as well as the multiscale numerical

indicator J multih (amacro, b) simultaneously, that is

bmeso = arg minb∈Bmeso

J meso(b), (22)

where J meso(b) is the (vector-valued) multi-objective cost function defined for any vector b ∈ Bmeso

byJ meso(b) =

(J meso

δ (b),J meso` (b),J multi

h (amacro, b))

. (23)

8. Numerical Methods for Solving the Multi-Objective Optimization Problem

The deterministic boundary value problem (2)-(6) defined on domain Ωmacro at macroscale and thestochastic boundary value problem (7)-(10) defined on a subdomain Ωmeso ⊂ Ωmacro at mesoscale are bothdiscretized using a classical displacement-based finite element method (FEM) [157,158]. The mathematicalexpectations of the quantities of interest of the stochastic boundary value problem (7)-(10) involved in thethree numerical indicators J meso

δ (b), J meso` (b) and J multi

h (amacro, b) are estimated using the Monte Carlonumerical simulation method [106–108,159,160] with Ns independent realizations Cmeso(θr)16r6Ns

of Cmeso. For the computation of the optimal value amacro, the classical single-objective optimizationproblem (21) is solved using the Nelder-Mead simplex algorithm [161–165]. For the computation of theoptimal value bmeso, the non-trivial multi-objective optimization problem (22) does not admit a singleglobal optimal solution, but inherently gives rise to a set of optimal solutions (called Pareto optima)resulting from a trade-off among the three components J meso

δ (b), J meso` (b) and J multi

h (amacro, b) ofthe multi-objective cost function J meso(b) which are competing and a priori conflicting. Based on theconcept of noninferiority [166] (also called Pareto optimality) for characterizing the components of amulti-objective function, a noninferior (or Pareto optimal) solution is such that an improvement in anyobjective function requires a degradation of some of the other objective functions, whereas an inferiorsolution is such that an improvement can be attained in all the objective functions. The set of all thenoninferior solutions in the parameter space is called the Pareto optimal set and the correspondingobjective function values in the multidimensional objective function space is called the Pareto optimalfront. The interested reader can refer to References [151–156,167] and the references therein for an overviewof nonlinear multi-objective optimization methods including the fundamental principles, some Pareto(near-)optimality conditions and a number of traditional and evolutionary optimization algorithms. InReference [140], the multi-objective optimization problem under consideration has been successfullysolved by using the genetic algorithm [151,156] that allows for constructing and finding a set of local

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Pareto optimal solutions that should be sufficiently representative of the whole Pareto optimal set andas many and diverse as possible for further selection [153,167]. The best compromise optimal solutionis selected among all the potential Pareto optimal solutions as the one that minimizes the distance to autopian solution that is constituted by the individual optimal solutions of the conflicting components ofthe multi-objective function, which corresponds to the origin of the Pareto front.

In the present work, a dedicated numerical indicator has been set up specifically for each componentof hyperparameter b = (δ, `, h), allowing for the use of a simpler and more efficient multi-objectiveoptimization algorithm, namely a fixed-point iterative algorithm. Starting from an ad hoc initial guess, itconsists in sequentially minimizing J meso

δ (b), J meso` (b) and J multi

h (amacro, b) respectively with respect toδ, ` and h in their sets of admissible values that are such that b = (δ, `, h) belongs to Bmeso. The iterativeprocess is stopped when the residual norm between two iterates becomes lower than a user-specifiedprescribed tolerance for each of the three single-objective optimization problems. Numerical results haveshown that, for the problem under consideration, such a fixed-point iterative algorithm can achieve thesame precision as the genetic algorithm in terms of convergence but with a lower overall computational cost(see the numerical examples in Sections 10 and 11). The main drawback of such a numerical optimizationalgorithm lies in the choice of the initial values used to start the algorithm that may be critical for thelocalization of the final global convergence region. Besides, note that the fixed-point iterative methodintroduced in this work could a priori be applied to the original formulation proposed in Reference [140],but it would lead to minimize the objective function J meso

δ (b) with respect to δ and ` simultaneouslygiven the other hyperparameters h. Although it is possible, the problem is that J meso

δ (b) is very sensitiveto δ but less sensitive with respect to `, since it has been tailored to perform the identification of the optimalvalue δmeso of δ and not the one `meso of `. Consequently, using such a fixed-point iterative strategy wouldyield uncertainties on the identified value `meso of `. It is the reason why the additional objective functionJ meso` (b) has been introduced and for which the sensitivity is of first order with respect to ` and of second

order with respect to δ.

9. Probabilistic Model for a Robust Identification of the Hyperparameters

When several non-overlapping mesoscopic domains of observation Ωmesoexp,1, . . . , Ωmeso

exp,Q are availablefor experimental measurements for the same test specimen instead of a unique observation domainΩmeso

exp , then the solution of the multi-objective optimization problem presented in Section 7 can yielddifferent optimal values bmeso

1 , . . . , bmesoQ of hyperparameter bmeso when experimental data comes from

one mesoscopic domain of observation to another since mesoscopic indicators J mesoδ (b) and J meso

` (b)depend on the values of experimental displacement fields umeso

exp,1, . . . , umesoexp,Q that are measured on each

of them. Consequently, the optimal value bmeso of hyperparameter b should be considered as uncertainand should be modeled as a vector-valued random variable B = (D, L, H) for which bmeso

1 , . . . , bmesoQ

are assumed to be Q independent realizations. Thus, in Reference [140], a robust identification of theoptimal value bopt is proposed by averaging the identified values bmeso

1 , . . . , bmesoQ . Nevertheless, in the

present work, an improved strategy is proposed that consists in constructing a prior stochastic modelof the vector-valued hyperparameter B by using the MaxEnt principle [68,72,73,77] and the availableinformation allowing for the explicit construction and parametric representation of the probability densityfunction pB : b 7→ pB(b) of random vector B. A robust identified value bopt is finally obtained usingthe MLE method [68–71] with the independent realizations bmeso

1 , . . . , bmesoQ . The available information

for constructing the prior stochastic model of B is as follows: (i) random variables D, L and H aremutually statistically independent, (ii) random variable D takes its values a.s. in ]0 , δsup[ with δsup =√(n + 1)/(n + 5) =

√7/11 ≈ 0.7977 < 1 (with n = 6 in linear elasticity), (iii) the random components of

random vector L are (statistically independent) positive-valued random variables a.s. for which the mean

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value is given in ]0 ,+∞[ and the values are unlikely near zero by construction of the mesoscale modeling,otherwise it would mean the current scale of the computational model is not correct and too large, (iv) therandom components of random vector H take their values a.s. in the admissible setHmeso. We then havefor all b = (δ, `, h) ∈ Bmeso,

pB(b) = pD(δ) pL(`) pH(h), (24)

wherepD(δ) =

1δsup

1]0,δsup[(δ), (25)

pL(`) =3

∏α=1

pLα(`α) with pLα(`α) = 1]0,+∞[(`α)1

bαaα Γ(aα)

`αaα−1 exp(−`α/bα). (26)

in which 1]0,δsup[ is the indicator function of the interval ]0 , δsup[ such that 1]0,δsup[(δ) = 1 if δ ∈ ]0 , δsup[

and 1]0,δsup[(δ) = 0 if δ 6∈ ]0 , δsup[, where s1 = (a1, b1), s2 = (a2, b2), s3 = (a3, b3) are positive parametersto be identified. We refer the reader to Reference [168] for a detailed construction of the prior stochasticmodel of H and a rigorous characterization of the statistical dependence between the components ofrandom elasticity tensors exhibiting a.s. some given material symmetry properties for the six highest levelsof linear elastic symmetries. For the special case of isotropic materials, we haveHmeso = ]0 ,+∞[×]0 ,+∞[

and the prior probability density function pH of random vector H is written as for all h = (h1, h2) ∈ Hmeso,

pH(h) = pH1(h1)×pH2(h2), (27)

in which

pH1(h1) = 1R+(h1)k1h1−λ exp (−λ1h1) , (28)

pH2(h2) = 1R+(h2)k2h2−5λ exp (−λ2h2) , (29)

where k1 = λ11−λ/Γ(1− λ) and k2 = λ2

1−5λ/Γ(1− 5λ) are two positive normalization constants. Theprobabilistic model of H is then parameterized by the vector-valued hyperparameter s = (λ, λ1, λ2) ∈]−∞ , 1/5[×]0 ,+∞[2. The mean values of H1 and H2 are respectively equal to (1−λ)/λ1 and (1− 5λ)/λ2,and the dispersion coefficients of H1 and H2 are respectively equal to 1/

√1− λ and 1/

√1− 5λ. Note

that the probability density functions of H1 and H2 both involve the same hyperparameter λ < 1/5that controls the level of statistical fluctuations of both H1 and H2. In addition, H1 and H2 cannot bedeterministic variables, since their dispersion coefficients are non zero whatever the value of λ < 1/5.Finally, the probabilistic model of B = (D, L, H) involves the unknown vector-valued hyperparameters = (s1, s2, s3, s) = (a1, b1, a2, b2, a3, b3, λ, λ1, λ2) belonging to the admissible set S = (]0 ,+∞[2)3×]−∞ , 1/5[×]0 ,+∞[2. The optimal value sopt of s is determined using the MLE method with the availabledata that are the Q independent realizations bmeso

1 , . . . , bmesoQ of random vector B. The MLE method

consists in computing sopt by solving the following optimization problem

sopt = arg maxs∈S

L(s; bmeso1 , . . . , bmeso

Q ), (30)

where s 7→ L(s; bmeso1 , . . . , bmeso

Q ) is the log-likelihood function for the Q independent realizationsbmeso

1 , . . . , bmesoQ of B which is defined for all s ∈ S by

L(s; bmeso1 , . . . , bmeso

Q ) =Q

∑q=1

log(pB(bmesoq ; s)). (31)

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The accuracy of the identified optimal value sopt is then all the higher as the number Q of mesoscopicdomains of observation is large but at the expense of a higher computational cost. Lastly, the optimalvalue bopt of vector-valued hyperparameter b ∈ Bmeso is computed by solving the following optimizationproblem

bopt = arg maxb∈Bmeso

pB(b; sopt). (32)

Hence, optimal value bopt corresponds to the most probable value of random vector B according to theidentified probability distribution represented by its probability density function pB(·; sopt) parameterizedby sopt. Note that the averaging approach presented in Reference [140] is a particular case of the MLEmethod presented in this section if the prior stochastic models of D, L and H are uniform random variables.It is the reason why a better robust identification is expected since the prior stochastic model of B has beenimproved in this work. In the present work, since D is modeled as a uniform random variable on ]0 , δsup[,the optimal value δopt of δ is simply obtained by averaging the Q independent realizations δmeso

1 , . . . , δmesoQ

of D. A more advanced prior stochastic model for D could have been considered, for instance by addingas available information that its mean value is given and its values are unlikely near zero, thus leading to aunimodal probability density function pD with support ]0 , δsup[ and with a higher parameterization thanthe simple uniform probability density function considered here.

10. Numerical Validation of the Multiscale Identification Method on In Silico Materials in 2D PlaneStress and 3D Linear Elasticity

In this section, we present a numerical application of the improved multiscale identificationmethodology proposed in the present work within the framework of 2D plane stress and 3D linearelasticity theories by using in silico materials for which the macroscopic and mesoscopic mechanicalproperties are known. The required multiscale “experimental” kinematic fields have been obtainedthrough numerical simulations using one random realization of the random elasticity field in SFE+ (seeSection 4) not restricted from R3 to some mesoscopic domain Ωmeso but restricted to the whole macroscopicdomain Ωmacro for a given experimental value bmeso

exp of hyperparameter b ∈ Bmeso. The solution of adeterministic boundary value problem over this macroscopic domain Ωmacro is then computed for aheterogeneous random elasticity field whose spatial correlation lengths correspond to the characteristicsizes of the heterogeneities at microscale. This deterministic boundary value problem is solved using aclassical numerical method (FEM) whose computational cost is high and potentially prohibitive in 3D,what can be avoided by computational homogenization methods, but it is required to completely simulatethe multiscale “experimental” measurements.

10.1. Validation on an In Silico Specimen in Compression Test in 2D Plane Stress Linear Elasticity

For this first numerical validation example, a 2D plane stress assumption is considered. Macroscopicdomain of observation Ωmacro

exp is a 2D square domain and it exactly corresponds to the cross-sectionof macroscopic domain Ωmacro and such that Ωmacro

obs = Ωmacroexp since the test specimen is in silico. The

dimensions of 2D macroscopic domain of observation Ωmacroexp are 1×1 cm2 in a fixed Cartesian frame

(O, x1, x2) of R2. It is possible to introduce a set of Q = 16 non-overlapping 2D square mesoscopic domainsof observation Ωmeso

exp,1, . . . , Ωmesoexp,Q ⊂ Ωmacro

exp for which the mesoscale dimensions are 1×1 mm2 (see Figure 1for a schematic representation of domains of observation Ωmacro

exp and Ωmesoexp,1, . . . , Ωmeso

exp,Q). Consequently,

observation domain Ωmesoobs , for which the dimensions are also 1×1 mm2, is defined as the 2D square

cross-section of mesoscopic domain Ωmeso. Deterministic surface force field f macro is uniformly distributedon the top boundary of macroscopic domain Ωmacro and applied along the (downward vertical) −x2

20 of 43

direction with an intensity of 5 kN such that ‖ f macro‖ = 5 kN/cm2 = 5×107 N/m2, while the bottomboundary of macroscopic domain Ωmacro is clamped.

10.1.1. Parameterization of the Macroscopic and Mesoscopic Models

At macroscale, the solution of deterministic boundary value problem (2)-(6) with 2D plane stressassumption depends only on 6 components Smacro(a)ijkh of deterministic compliance tensor Smacro(a)with i, j, k, h ∈ 1, 2. Consequently, the solution at macroscale depends only on the components of a2D fourth-order compliance tensor Smacro

2D (a) that is defined by Smacro2D (a)ijkh = Smacro(a)ijkh for all

i, j, k, h ∈ 1, 2. Then, a 2D fourth-order elasticity tensor at macroscale can be introduced and definedby Cmacro

2D (a) = (Smacro2D (a))−1. Since within the framework of linear elasticity theory, any isotropic

material is completely characterized by a bulk modulus κ and a shear modulus µ at macroscale, thenwe have the vector-valued parameter a = (κ, µ). In particular, we have chosen the experimental valueamacro

exp = (κmacroexp , µmacro

exp ) with κmacroexp = 13.901 GPa and µmacro

exp = 3.685 GPa, corresponding to a Young’smodulus Emacro

exp = 10.158 GPa and and a Poisson’s ratio νmacroexp = 0.3782.

At mesoscale, the solution of stochastic boundary value problem (7)-(10) with 2D plane stressassumption depends only on 6 components Smeso(b)ijkh of random compliance tensor field Smeso(b)with i, j, k, h ∈ 1, 2 or equivalently on every 21 components Cmeso(b)ijkh of random elasticity tensorfield Cmeso(b) with i, j, k, h ∈ 1, 2, 3. It is the reason why we have chosen to construct the prior stochasticmodel of the random compliance tensor field Smeso(b) as presented in Section 4 and the stochasticboundary value problem (7)-(10) is solved in using (11) rather than (8). Furthermore, its mean functionSmeso is spatially constant and models an isotropic elastic medium that is completely characterized bya mean bulk modulus κ and a mean shear modulus µ at mesoscale. Consequently, the vector-valuedhyperparameter b = (δ, `, h) involves only (i) a dispersion parameter δ, (ii) a spatial correlation length` that is such that `1 = `2 = ` in order to be consistent with the effective model at macroscale forwhich the material is assumed to be isotropic and with `3 = +∞ in order to be consistent with the 2Dplane stress assumption, and (iii) a vector-valued hyperparameter h = (κ, µ) gathering the mean bulkmodulus κ and the mean shear modulus µ at mesoscale. In particular, we have chosen the experimentalvalue bmeso

exp = (δmesoexp , `meso

exp , κmesoexp , µmeso

exp) with δmeso

exp = 0.40, `mesoexp = 125 µm, κmeso

exp = 13.75 GPa and

µmesoexp

= 3.587 GPa, corresponding to a mean Young’s modulus Emesoexp = 9.900 GPa and a mean Poisson’s

ratio νmesoexp = 0.380 GPa. For identification purposes and further computational savings, we consider a

reduced admissible set Bmesoad ⊂ Bmeso for the vector-valued hyperparameter b = (δ, `, κ, µ) such that

δ ∈ [0.25 , 0.50], ` ∈ [20 , 250] µm, κ ∈ [8.5 , 17] GPa, µ ∈ [2.15 , 4.50] GPa, instead of the full admissible set

Bmeso = ]0 , δsup[×]0 ,+∞[×]0 ,+∞[2 with δsup =√(n + 1)/(n + 5) =

√7/11 ≈ 0.7977 < 1 (with n = 6

in linear elasticity). This reduced admissible set Bmesoad is then discretized into nV = 10 equidistant points in

each dimension for which the three numerical indicators J mesoδ (b), J meso

` (b) and J multih (amacro, b) defined

in Section 6.4 are evaluated and compared. The identified values bmeso1 , . . . , bmeso

Q of hyperparametersb for each of the Q mesoscopic domains of observation Ωmeso

exp,1, . . . , Ωmesoexp,Q are then searched on this

multidimensional grid of nV×nV×nV×nV points in the hypercube Bmesoad .

Within the framework of linear elasticity under 2D plane stress assumption, both the deterministicboundary value problem (2)-(6) and the stochastic boundary value problem (7)-(10) are solved bydiscretizing the 2D macroscopic and mesoscopic domains of observation Ωmacro

obs and Ωmesoobs in space

using the FEM. The finite element meshes of 2D square domains Ωmacroobs and Ωmeso

obs are structured meshesmade up with 4-nodes linear quadrangular elements with Gauss-Legendre quadrature rule. The stochasticboundary value problem (7)-(10) at mesoscale is solved using the Monte Carlo numerical method. Meshconvergence analyses of the numerical solutions of the deterministic boundary value problem (2)-(6) atmacroscale and of the stochastic boundary value problem (7)-(10) at mesoscale have been performed in

21 of 43

order to define accurate finite element approximations at both macroscopic and mesoscopic scales. Thefinite element mesh of 2D macroscopic domain Ωmacro

obs is a regular grid containing 25×25 quadrangularelements with uniform element size hmacro = 0.4 mm = 4×10−4 m in each spatial direction. It thuscomprises 676 nodes and 625 elements, with 1300 unknown degrees of freedom (dofs). The finite elementmesh of 2D mesoscopic domain Ωmeso

obs is a regular grid containing 100×100 quadrangular elements withuniform element size hmeso = 10 µm = 10−5 m in each spatial direction. It thus comprises 10, 201 nodesand 10, 000 elements, with 20, 000 unknown dofs. The number of Gauss integration points per spatialcorrelation length used for numerical quadrature over 2D macroscopic domain of observation Ωmacro

obs and2D mesoscopic domain of observation Ωmeso

obs is nG = 4 in each spatial direction.Concerning the computational stochastic homogenization with 2D plane stress assumption,

we consider a 2D square domain ΩRVE of side length BRVE defined in a Cartesian frame (O, x1, x2) and weuse the homogenization method with static uniform boundary conditions (i.e., with homogeneous stresses)which is appropriate for linear elasticity under 2D plane stress assumption. Note that only the componentsSeff(b)ijkh with i, j, k, h ∈ 1, 2 can be calculated. We then obtain a 2D fourth-order effective compliancetensor Seff

2D(b) that is such that Seff2D(b)ijkh = Seff(b)ijkh for all i, j, k, h ∈ 1, 2. Then, a 2D fourth-order

effective elasticity tensor can be defined as Ceff2D(b) = (Seff

2D(b))−1. A convergence analysis of the statistical

estimator of its statistical fluctuations with respect to the representative volume element size BRVE hasbeen performed. A representative volume element size BRVE = 20×` = 400 µm = 4×10−4 m has beenfound to be sufficient to reach negligible statistical fluctuations for the construction of the multiscalenumerical indicator J multi

h (amacro, b) that is calculated by replacing Cmacro(a) and Ceff(b) with Cmacro2D (a)

and Ceff2D(b), respectively, in (18).

As the mathematical expectations involved in each of the numerical indicators J mesoδ (b), J meso

` (b)and J multi

h (amacro, b) are evaluated using the Monte Carlo numerical method, statistical convergenceanalyses of their statistical estimators with respect to the number of independent realizations Ns havebeen carried out and a convergence has been reached for Ns = 500. Sensitivity analyses of each ofthe three numerical indicators have been performed with respect to each of the hyperparameters δ, `,h = (κ, µ), respectively, in the reduced admissible set Bmeso

ad = [0.25 , 0.50]×[20 , 250] µm×[8.5 , 17] GPa×[2.15 , 4.50] GPa. Hence, it can be shown that each numerical indicator is sufficiently sensitive to thevariation of its dedicated hyperparameter and that the multi-objective optimization problem (22) to besolved is well-posed.

Recall the multiscale statistical inverse problem has been formulated into two decoupled optimizationproblems in a and b, respectively, to be solved sequentially (see Section 7), namely (i) a macroscalesingle-objective optimization problem (21) for the inverse identification of the optimal value amacro ofparameter a in its admissible set Amacro, and (ii) a mesoscale multi-objective optimization problem (22) forthe statistical inverse identification of the global optimal value bopt of hyperparameter b in its reducedadmissible set Bmeso

ad .

10.1.2. Resolution of the Single-Objective Optimization Problem at Macroscale

In this paragraph, we present the results of the first single-objective optimization problem (21) atmacroscale which consists in minimizing the macroscopic numerical indicator J macro(a) constructed in themacroscopic domain of observation Ωmacro

exp for identifying the optimal value amacro of a at macroscale. Thesingle-objective optimization problem (21) at macroscale has been solved using the Nelder-Mead simplexalgorithm. The identification results are reported in Table 1 and show that the relative error betweenthe identified optimal value amacro = (13.901, 3.685) in [GPa] and the reference experimental valueamacro

exp = (14.328, 3.670) in [GPa] used for the construction of the numerically simulated “experimental”database remains small (less than 3% and 0.5% for κmacro and µmacro, respectively), allowing to validate

22 of 43

the proposed identification methodology in 2D plane stress linear elasticity for the resolution of thesingle-objective optimization problem (21) at macroscale.

Table 1. Comparison between the identified optimal value amacro and the reference experimentalvalue amacro

exp .

κ [GPa] µ [GPa]

amacro 13.901 3.685amacro

exp 14.328 3.670Relative error [%] 2.980 0.4009

10.1.3. Resolution of the Multi-Objective Optimization Problem at Mesoscale

In this paragraph, we present the results of the second multi-objective optimization problem (22)at mesoscale which consists in simultaneously minimizing the three numerical indicators J meso

δ (b),J meso` (b) and J multi

h (amacro, b) constructed in each of the Q = 16 mesoscopic domains of observationΩmeso

exp,1, . . . , Ωmesoexp,Q using the optimal parameter amacro = (13.901, 3.685) in [GPa] previously identified at

macroscale (see the previous paragraph) for identifying the global optimal value bopt of b at mesoscale.The multi-objective optimization problem (22) at mesoscale has been solved using the fixed-point iterativealgorithm on the one hand and the genetic algorithm on the other hand for comparison purposes. In orderto analyze the numerical efficiency of these two resolution approaches, instead of evaluating the computingtime which strongly depends on the computer hardware used, we choose in this work to compare thenumber of evaluations of the random solution of the stochastic boundary value problem (7)-(10) atmesoscale (i.e., the number of calls to the deterministic numerical model at mesoscale) required by eachalgorithm to achieve the desired convergence.

The identification results obtained with the fixed-point iterative algorithm are summarized in Table 2for the set of Q = 16 mesoscopic domains of observation Ωmeso

exp,1, . . . , Ωmesoexp,Q, namely the set of Q identified

values bmeso1 , . . . , bmeso

Q and numbers of iterations n1, . . . , nQ required to reach the desired convergence,with a convergence criterion on the residual norm between two iterations that must be less than a prescribedtolerance set to 10−9, and the global optimal value bopt computed by using the MLE method. On theone hand, there are greater variations between the identified values `meso

1 , . . . , `mesoQ and δmeso

1 , . . . , δmesoQ ,

reflecting the fact that the two associated mesoscopic numerical indicators J mesoδ (b) and J meso

` (b) dependdirectly on the experimental field measurements on each mesoscopic domain of observation. On theother hand, the lower variability between the identified values κmeso

1 , . . . , κmesoQ and µmeso

1, . . . , µmeso

Qcan

be explained by the fact that the associated multiscale numerical indicator J multih (amacro, b) does not

depend directly on the experimental field measurements on each mesoscopic domain of observation but israther conditioned by the identified values `meso

1 , . . . , `mesoQ and δmeso

1 , . . . , δmesoQ . Thus, the relative errors

calculated on these two hyperparameters are essentially due to the quality of the discretization of thereduced admissible set Bmeso

ad . In particular, the fixed-point iterative algorithm has selected the sameidentified value µmeso

1= · · · = µmeso

Q= 3.717 GPa (among the nV = 10 test points in [2.15 , 4.50] GPa) for

the Q = 16 mesoscopic domains of observation Ωmesoexp,1, . . . , Ωmeso

exp,Q. Clearly, a finer grid (with nV > 10)might yield different values for the identified hyperparameter µmeso selected by the optimization algorithm.It is the reason why a prior probabilistic model for the identified hyperparameters has been introduced.The number of evaluations of the stochastic computational model needed by the fixed-point iterativealgorithm is given by nFP

tot = 3 nV Ns ∑16q=1 nq, where the superscript FP refers to “Fixed-Point” and nV

is the number of evaluations of a numerical indicator to search for the minimum with respect to theassociated hyperparameter. Figure 3 shows the probability density functions pD, pL, pK and pM of randomvariables D, L, K and M, respectively, which are defined in Section 9 with the two components H1 = K

23 of 43

and H2 = M of random vector H = (K, M). As suggested by the identification results shown in Table 2and as already mentioned in Section 9, a more advanced prior stochastic model for D would have beenpreferable to obtain a unimodal probability density function pD with support ]0 , δsup[ and which would beconcentrated around the reference experimental value δmeso

exp = 0.4. Besides, although all the independentrealizations µmeso

1, . . . , µmeso

Qof M given in Table 2 are equal to the same identified value 3.717 GPa, the

probability density function pM does not correspond to the Dirac measure on R at point 3.717 GPa but to agamma distribution with a very small dispersion around this value, since for the prior probabilistic modelof H = (K, M) considered here, K and M cannot be deterministic variables (see Section 9). We finallyobtain the global optimal value bopt = (0.391, 135.328, 12.273, 3.717) in ([−], [µm], [GPa], [GPa]) withrelative errors less than 3%, 9%, 11% and 4% for δopt, `opt, κopt and µopt, respectively, with respect to thereference experimental value bmeso

exp = (0.40, 125, 13.75, 3.587) in ([−], [µm], [GPa], [GPa]) used to constructthe numerically simulated “experimental” database, allowing to validate the proposed identificationmethodology in 2D plane stress linear elasticity for the resolution of the multi-objective optimizationproblem (22) at mesoscale.

Table 2. Fixed-point iterative algorithm: comparison between the global optimal value bopt obtainedfrom the Q = 16 identified values bmeso

1 , . . . , bmesoQ for each of the Q mesoscopic domains of observation

Ωmesoexp,1, . . . , Ωmeso

exp,Q, and the reference experimental value bmesoexp .

δ ` [µm] κ [GPa] µ [GPa] nq

bmeso1 0.306 147.778 13.222 3.717 3

bmeso2 0.500 224.444 11.333 3.717 4

bmeso3 0.417 122.222 12.278 3.717 3

bmeso4 0.417 122.222 12.278 3.717 3

bmeso5 0.444 147.778 12.278 3.717 3

bmeso6 0.417 122.222 12.278 3.717 4

bmeso7 0.361 147.778 12.278 3.717 4

bmeso8 0.361 147.778 12.278 3.717 4

bmeso9 0.444 147.778 12.278 3.717 3

bmeso10 0.333 147.778 12.278 3.717 4

bmeso11 0.333 122.222 12.278 3.717 4

bmeso12 0.389 96.667 12.278 3.717 3

bmeso13 0.389 147.778 12.278 3.717 4

bmeso14 0.389 122.222 12.278 3.717 3

bmeso15 0.389 147.778 12.278 3.717 4

bmeso16 0.361 122.222 12.278 3.717 4

bopt 0.391 135.328 12.273 3.717 -bmeso

exp 0.400 125.000 13.750 3.587 -Relative error [%] 2.344 8.262 10.740 3.611 -

nFPtot 855, 000

24 of 43

0 0.2 0.4 0.60

0.51

1.52

2.5

δ

p D(δ)

(a)

0.5 1 1.5 2 2.5

×10−4

0

0.5

1

1.5

×104

` [m]

p L(`)

(b)

10 12 14 160

0.5

1

1.5

×10−9

κ [GPa]

p K(κ)

(c)

2.5 3 3.5 4 4.50

0.20.40.60.8

11.21.4×10−8

µ [GPa]

p M(µ)

(d)Figure 3. Fixed-point iterative algorithm: probability density functions pD, pL, pK and pM of randomvariables D, L, K and M, respectively. (a) pD(δ); (b) pL(`); (c) pK(κ); (d) pM(µ).

Figure 4 shows the evolution of the global optimal values δopt, `opt, κopt µopt estimated by the MLEmethod as a function of the number Q of independent realizations bmeso

1 , . . . , bmesoQ of random vector

B = (D, L, K, M). Although the number Q remains low (less than or equal to 16), we observe that eachof the global optimal values tends to converge towards an objective value when Q increases, whichdemonstrates that the use of the MLE method with the prior probabilistic model of B proposed in thiswork allows a robust identification of the vector-valued hyperparameter b = (δ, `, κ, µ).

2 4 6 8 101214160.25

0.30.35

0.40.45

0.5

Q

δopt

(a)

2 4 6 8 101214161

1.21.41.61.8

2×10−4

Q

`opt[m

]

(b)

2 4 6 8 1012141611

11.512

12.513

13.514

Q

κop

t[G

Pa]

(c)

2 4 6 8 101214163.71

3.71

3.72

3.72

3.72

Q

µop

t[G

Pa]

(d)

Figure 4. Fixed-point iterative algorithm: evolutions of the identified global optimal values δopt, `opt, κopt

and µopt with respect to the number Q of mesoscopic domains of observation considered. (a) δopt(Q); (b)`opt(Q); (c) κopt(Q); (d) µopt(Q).

In terms of computational efficiency, we can see in Table 2 that the numbers of iterations n1, . . . , nQrequired to achieve the desired convergence are relatively low (less than or equal to 4) for each ofthe Q = 16 mesoscopic domains of observation Ωmeso

exp,1, . . . , Ωmesoexp,Q, leading to a number of calls to the

deterministic numerical model at mesoscale of 855, 000. Table 3 contains the global optimal values bopt

and the corresponding relative errors (with respect to the reference experimental value bmesoexp ) obtained for

different values Ns ∈ 5, 50, 500 of the number of independent realizations generated for the statisticalestimation of the mathematical expectations involved in the different numerical indicators. It can beseen that a strong decrease in the value of Ns allows a considerable gain in computing time whilemaintaining similar results for the identified global optimal values, which can be explained by the useof the MLE method which makes the resolution of the statistical inverse identification problem morerobust with respect to the convergence of the statistical estimators used in the numerical indicators of themulti-objective optimization problem (22).

25 of 43

Table 3. Fixed-point iterative algorithm: comparison between the global optimal value bopt and thereference experimental value bmeso

exp for different values of the number Ns of independent realizationsgenerated for the statistical estimation of the mathematical expectations involved in the different numericalindicators.

δ ` [µm] κ [GPa] µ [GPa] nFPtot

bmesoexp 0.400 125.000 13.750 3.587 -

bopt (Ns = 500) 0.391 135.328 12.273 3.717 855, 000Relative error [%] 2.344 8.262 10.740 3.611 -

bopt (Ns = 50) 0.387 134.859 12.217 3.717 87, 000Relative error [%] 3.212 7.887 11.153 3.611 -

bopt (Ns = 5) 0.396 140.220 12.335 3.717 9, 000Relative error [%] 1.042 12.176 10.293 3.611 -

The identification results obtained with the genetic algorithm are summarized in Table 4 for the setof Q = 16 mesoscopic domains of observation Ωmeso

exp,1, . . . , Ωmesoexp,Q, namely the set of Q identified values

bmeso1 , . . . , bmeso

Q and numbers of generations n1, . . . , nQ required to reach the desired convergence, and theglobal optimal value bopt computed by using the MLE method. The initial population used to start thegenetic algorithm contains nI = 40 individuals. Figure 5 shows an example of different 2D cross-sectionsof the Pareto front for the first mesoscopic domain of observation Ωmeso

exp,1. The best comprise optimalsolution corresponds to the point marked with a green circle on the different 2D cross-sections of the Paretofront, because according to the explanations given in Section 8, it is chosen among all the noninferior(Pareto optimal) solutions generated and selected in the optimal Pareto set (represented by red stars inFigure 5) as the one that minimizes the distance at the origin of the Pareto front in the multidimensionalspace (of dimension 3) of the multi-objective cost function J meso(b). For reasons of limitation in termsof calculation cost, the number Ns of independent realizations used for the statistical estimation of themathematical expectations involved in the numerical indicators J meso

δ (b), J meso` (b) and J multi

h (amacro, b)is reduced to Ns = 50. Although the statistical convergence of the three numerical indicators is notachieved, the results of Table 3 show that, thanks to the probabilistic modeling of the hyperparametersand the maximum likelihood estimation, the results of the statistical inverse identification method arenot significantly affected by a decrease in the value of Ns and are therefore robust with respect to thestatistical fluctuations of the different numerical indicators. The number of evaluations of the stochasticcomputational model needed by the genetic algorithm is given by nGA

tot = 3 nI Ns ∑16q=1 nq, where the

superscript GA refers to “Genetic Algorithm”. Figure 6 shows the probability density functions pD, pL,pK and pM of random variables D, L, K and M, respectively. We finally deduce the global optimal valuebopt = (0.372, 128.401, 11.656, 3.306) in ([−], [µm], [GPa], [GPa]) with relative errors less than 8%, 3%,16% and 8% for δopt, `opt, κopt and µopt, respectively, with respect to the reference experimental valuebmeso

exp = (0.40, 125, 13.75, 3.587) in ([−], [µm], [GPa], [GPa]), which are acceptable (reasonably good) andsimilar to the errors obtained by the fixed-point iterative algorithm. There are still some fluctuations inthe values κmeso

1 , . . . , κmesoQ and µmeso

1, . . . , µmeso

Qidentified on each of the Q = 16 mesoscopic domains

of observation Ωmesoexp,1, . . . , Ωmeso

exp,Q, which was not the case for the fixed-point iterative algorithm. Thisunderlies the numerical resolution of the Pareto front, which depends on the number nV of values in eachdimension of the parameter search space. In terms of computational efficiency, we can see that the numbernGA

tot = 19, 176, 000 of evaluations of the stochastic computational model (resulting from the number ofindividuals nI = 40 in the initial population and the number of population generations n1, . . . , nQ) ismuch higher than that nFP

tot = 87, 000 required by the fixed-point iterative algorithm with Ns = 50 (see

26 of 43

Table 3). Finally, the fixed-point iterative algorithm allows significant computational savings (in terms ofcomputational cost) compared to the genetic algorithm.

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

J mesoδ (b)

Jm

eso

`(b)

(a)

0 0.1 0.2 0.3 0.4

0

0.05

0.1

0.15

0.2

J mesoδ (b)

Jm

ulti

h(a

mac

ro,b)

(b)

0 0.1 0.2 0.3 0.4

0

0.05

0.1

0.15

0.2

J meso` (b)

Jm

ulti

h(a

mac

ro,b)

(c)Figure 5. Different 2D cross-sections of the Pareto front with the noninferior (Pareto optimal) solutionsrepresented by red stars ? and the best compromise optimal solution surrounded by a green circle # forthe mesoscopic domain of observation Ωmeso

exp,1. (a) cross-section (Jmesoδ (b),Jmeso

` (b)); (b) cross-section

(Jmesoδ (b),Jmulti

h (amacro, b)); (c) cross-section (Jmeso` (b),Jmulti

h (amacro, b)).

Table 4. Genetic algorithm: comparison between the global optimal value bopt obtained from the Q = 16identified values bmeso

1 , . . . , bmesoQ for each of the Q mesoscopic domains of observation Ωmeso

exp,1, . . . , Ωmesoexp,Q,

and the reference experimental value bmesoexp .

δ ` [µm] κ [GPa] µ [GPa] nq

bmeso1 0.361 122.222 16.056 2.411 193

bmeso2 0.333 147.778 9.444 2.933 202

bmeso3 0.417 198.889 13.222 3.194 189

bmeso4 0.333 147.778 13.222 3.456 197

bmeso5 0.444 147.778 11.333 4.239 207

bmeso6 0.417 173.333 12.278 2.933 201

bmeso7 0.278 147.778 10.389 3.717 192

bmeso8 0.278 147.778 12.278 3.194 199

bmeso9 0.389 96.667 14.167 3.978 210

bmeso10 0.333 96.667 11.333 2.933 205

bmeso11 0.278 96.667 15.111 2.933 203

bmeso12 0.417 122.222 12.278 4.239 198

bmeso13 0.472 122.222 14.167 3.456 194

bmeso14 0.389 96.667 12.278 2.672 208

bmeso15 0.361 122.222 14.167 3.456 190

bmeso16 0.444 173.333 9.444 3.978 208

bopt 0.372 128.401 11.656 3.306 -bmeso

exp 0.400 125.000 13.750 3.587 -Relative error [%] 7.118 2.721 15.228 7.844 -

nGAtot 19, 176, 000

27 of 43

0 0.2 0.4 0.60

0.51

1.52

2.5

δ

p D(δ)

(a)

0.5 1 1.5 2 2.5

×10−4

00.20.40.60.8

11.21.4×104

` [m]

p L(`)

(b)

10 12 14 160.40.60.8

11.2×10−10

κ [GPa]

p K(κ)

(c)

2.5 3 3.5 4 4.50

0.20.40.60.8

1×10−9

µ [GPa]

p M(µ)

(d)Figure 6. Genetic algorithm: probability density functions pD, pL, pK and pM of random variables D, L, Kand M, respectively. (a) pD(δ); (b) pL(`); (c) pK(κ); (d) pM(µ).

10.2. Validation on an In Silico Specimen in Compression Test in 3D Linear Elasticity

In this section, we present a second validation example in 3D linear elasticity. We assume there areQ = 3 test specimens on which are applied exactly the same external loads at macroscale. Recall that for thevalidation, the “experimental” tests are actually performed in silico. Macroscopic domain of observationΩmacro

exp is exactly the same 3D cubic domain for each test specimen and corresponds to 3D experimentalfield measurements on the full volume of each test specimen. As for the previous 2D validation example,since the experimental field measurements are actually performed in silico, we also have Ωmacro

obs = Ωmacroexp .

The dimensions of each 3D macroscopic domain of observation Ωmacroexp are 2×2×2 mm3. For each test

specimen, the mesoscale dimensions of 3D mesoscopic domain of observation Ωmesoexp are 0.5×0.5×0.5 mm3

(see Figure 7). Deterministic surface force field f macro is uniformly distributed on the top boundary ofmacroscopic domain Ωmacro

exp and applied along the (downward vertical) −x3 direction with an intensityof 2 kN such that ‖ f macro‖ = 50 kN/cm2 = 5×108 N/m2, while the bottom boundary of macroscopicdomain Ωmacro

exp is clamped.

01

2

0

1

20

1

2

x1 [mm]x2 [mm]

x 3[m

m]

Figure 7. Illustration of the test specimen occupying the 3D cubic macroscopic domain of observationΩmacro

exp = Ωmacro (in green) which contains a 3D cubic mesoscopic domain of observation Ωmesoexp = Ωmeso

(in red) for the numerical validation in 3D linear elasticity.

10.2.1. Parameterization of the Macroscopic and Mesoscopic Models

Within the framework of linear elasticity theory, any material that is isotropic at macroscale canbe completely characterized by a bulk modulus κ and a shear modulus µ. Consequently, we havechosen the parameterization a = (κ, µ). In particular, we have chosen the experimental value amacro

exp =

(κmacroexp , µmacro

exp ) with κmacroexp = 138.783 GPa and µmacro

exp = 64.355 GPa, corresponding to a Young’s modulusEmacro

exp = 167.218 GPa and a Poisson’s ratio νmacroexp = 0.2992.

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At mesoscale, we have chosen to construct the prior stochastic model of the random elasticity tensorfield Cmeso as presented in Section 4 and the stochastic boundary value problem (7)-(10) is solved in using(8) rather than (11). Furthermore, its mean function Cmeso is spatially constant and models an isotropicelastic medium that is completely characterized by a mean bulk modulus κ and a mean shear modulus µ

at mesoscale. Consequently, the vector-valued hyperparameter b = (δ, `, h) involves only (i) a dispersionparameter δ, (ii) a spatial correlation length ` that is such that `1 = `2 = `3 = ` in order to be consistentwith the effective model at macroscale for which the material is assumed to be isotropic, and (iii) avector-valued hyperparameter h = (κ, µ) gathering the mean bulk modulus κ and the mean shear modulusµ at mesoscale. In particular, we have chosen the experimental value bmeso

exp = (δmesoexp , `meso

exp , κmesoexp , µmeso

exp)

with δmesoexp = 0.32, `meso

exp = 80 µm, κmesoexp = 145 GPa and µmeso

exp= 67.3 GPa, corresponding to a mean

Young’s modulus Emesoexp = 174.85 GPa and a mean Poisson’s ratio νmeso

exp = 0.2990 GPa. As already

mentioned in Section 10.1.1, we can restrict the admissible set Bmeso = ]0 , δsup[×]0 ,+∞[×]0 ,+∞[2 (withδsup =

√7/11 ≈ 0.7977 < 1) of the vector-valued hyperparameter b = (δ, `, κ, µ) to a reduced admissible

set Bmesoad ⊂ Bmeso such that δ ∈ [0.20 , 0.45], ` ∈ [50 , 120] µm, κ ∈ [87.5 , 200] GPa and µ ∈ [40.5 , 95.0] GPa.

This reduced admissible set Bmesoad is then discretized into nV = 10 equidistant points in each dimension

for which the three numerical indicators J mesoδ (b), J meso

` (b) and J multih (amacro, b) defined in Section 6.4

are evaluated and compared. The identified values bmeso1 , bmeso

2 , bmeso3 of hyperparameters b for each of

the 3 in silico test specimens are then searched on this multidimensional grid of nV×nV×nV×nV points inthe hypercube Bmeso

ad .The classical displacement-based FEM is used for the spatial discretization of (i) the deterministic

boundary value problems defined by (2)-(6) in replacing Cmacro by Q = 3 independent realizationsof the random apparent elasticity tensor field Cmeso on Ωmacro instead of Ωmeso to simulate boththe “experimental” macroscopic displacement field umacro

exp in Ωmacroexp = Ωmacro and the mesoscopic

displacement field umesoexp in Ωmeso

exp = Ωmeso, (ii) the deterministic boundary value problem defined by (2)-(6)to calculate the macroscopic displacement field umacro in domain Ωmacro

obs = Ωmacro, and (iii) the stochasticboundary value problems defined by (7)-(10) to calculate the random mesoscopic displacement fieldsUmeso in using experimental data obtained by solving (i) that are the experimental displacement fieldsumeso

exp measured on the boundary of domain Ωmesoexp = Ωmeso

obs for each realization of Cmeso. The stochasticsolver used for solving the stochastic boundary value problem (7)-(10) at mesoscale is the Monte Carlonumerical method. As 3D macroscopic and mesoscopic domains Ωmacro and Ωmeso are cubic domains,we consider for each of them a spatial discretization with a structured mesh made up with 8-nodes linearhexahedral elements with Gauss-Legendre quadrature rule. The finite element mesh of 3D macroscopicdomain Ωmacro is made with the same spatial discretization as the one used for the 2D validation exampleat macroscale, that is a structured mesh of 25×25×25 = 15, 625 hexahedral elements with uniformelement size hmacro = 80 µm = 8×10−5 m in each spatial direction. The finite element mesh of 3Dmesoscopic domain Ωmeso is made with the same spatial discretization as the one used for the 2Dvalidation example at mesoscale and whose element size depends on the smallest spatial correlation lengthconsidered, that is a structured mesh of 20×20×20 = 8000 hexahedral elements with uniform elementsize hmeso = `/(nG/2) = (50 µm)/2 = 25 µm = 2.5×10−5 m in each spatial direction, with nG = 4 Gaussintegration points per spatial correlation length.

Concerning the computational stochastic homogenization, as for the 2D validation example, the sizeBRVE of representative volume element ΩRVE is defined as a function of the spatial correlation length `

such that BRVE = 20×` = 20×50 µm = 1 mm = 10−3 m.Recall that the multiscale statistical inverse problem has been formulated into two decoupled

optimization problems in a and b, respectively, to be solved sequentially (see Section 7), namely (i)a macroscale single-objective optimization problem (21) for the inverse identification of the optimal value

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amacro of parameter a in its admissible set Amacro, and (ii) a mesoscale multi-objective optimizationproblem (22) for the statistical inverse identification of the global optimal value bopt of hyperparameter bin its reduced admissible set Bmeso

ad .

10.2.2. Resolution of the Single-Objective Optimization Problem at Macroscale

In this paragraph, we present the results of the first single-objective optimization problem (21) atmacroscale which consists in minimizing the macroscopic numerical indicator J macro(a) constructed ineach of the Q = 3 in silico test specimens for identifying the optimal value amacro of a at macroscale. Thesingle-objective optimization problem (21) at macroscale has been solved using the Nelder-Mead simplexalgorithm. The identification results are reported in Table 5 and show that the relative error betweenthe identified optimal value amacro = (138.783, 64.355) in [GPa] and the reference experimental valueamacro

exp = (138.758, 64.377) in [GPa] used for the construction of the numerically simulated “experimental”database remains very small (less than 0.02% and 0.04% for κmacro and µmacro, respectively), allowingto validate the proposed identification methodology in 3D linear elasticity for the resolution of thesingle-objective optimization problem (21) at macroscale.

Table 5. Comparison between the identified optimal value amacro and the reference experimentalvalue amacro

exp .

κ [GPa] µ [GPa]

amacro 138.783 64.355amacro

exp 138.758 64.377Relative error [%] 0.018 0.034

10.2.3. Resolution of the Multi-Objective Optimization Problem at Mesoscale

In this paragraph, we present the results of the second multi-objective optimization problem (22) atmesoscale which consists in simultaneously minimizing the three numerical indicators J meso

δ (b), J meso` (b)

and J multih (amacro, b) constructed in each of the Q = 3 in silico tests specimens using the optimal parameter

amacro = (138.783, 64.355) in [GPa] previously identified at macroscale (see the previous paragraph) foridentifying the global optimal value bopt of b at mesoscale.

In contrast, unlike the 2D validation example, the multi-objective optimization problem (22) hasbeen solved only with the fixed-point iterative algorithm using the same convergence criterion on theresidual norm between two iterations that must be less than a tolerance set to 10−9 and by searching for thesolution of the multi-objective optimization problem (22) in a multidimensional grid of nV×nV×nV×nVpoints in the reduced admissible set Bmeso

ad ⊂ R4. The genetic algorithm has not been used because theresulting computational cost was too high with the available computational resources. The number ofindependent realizations for the statistical estimation of the mathematical expectations involved in thedifferent numerical indicators is set to Ns = 500. The number of evaluations of the stochastic computationalmodel needed by the fixed-point iterative algorithm is given by nFP

tot = 3 nV Ns ∑3q=1 nq.

Table 6 reports the identification results obtained with the fixed-point iterative algorithm for the setof Q = 3 in silico tests specimens, namely the set of identified values bmeso

1 , bmeso2 , bmeso

3 and numbersof iterations n1, n2, n3 required to reach the desired convergence (with a tolerance set to 10−9), and theglobal optimal value bopt computed by using the MLE method. As for the 2D validation example, thereare more significant variations between the identified values `meso

1 , `meso2 , `meso

3 and δmeso1 , δmeso

2 , δmeso3 ,

again reflecting the fact that the two associated mesoscopic numerical indicators J mesoδ (b) and J meso

` (b)depend directly on the experimental field measurements on each in silico test specimen. The identifiedvalues κmeso

1 , κmeso2 , κmeso

3 and µmeso1

, µmeso2

, µmeso3

being almost identical for each in silico test specimen, we

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directly identify the global optimal values κopt and µopt without using the MLE method for the randomvariables K and M. Figure 8 shows the probability density functions pD and pL defined in Section 9and associated to random variables D and L, respectively. We finally obtain the global optimal valuebopt = (0.330, 91.236, 150.000, 64.722) in ([−], [µm], [GPa], [GPa]) with relative errors less than 4% forδopt, `opt, κopt and µopt with respect to the reference experimental value bmeso

exp = (0.32, 80, 145, 67.3) in([−], [µm], [GPa], [GPa]) used to construct the numerically simulated “experimental” database, allowingto validate the proposed identification methodology in 3D linear elasticity for the resolution of themulti-objective optimization problem (22) at mesoscale.

Table 6. Fixed-point iterative algorithm: comparison between the global optimal value bopt obtainedfrom the 3 identified values bmeso

1 , bmeso2 , bmeso

3 for each of the 3 in silico test specimens and the referenceexperimental value bmeso

exp .

δ ` [µm] κ [GPa] µ [GPa] nq

bmeso1 0.311 65.556 150.000 64.722 3

bmeso2 0.367 88.889 150.000 64.722 4

bmeso3 0.311 81.111 150.000 64.722 3

bopt 0.330 77.271 150.000 64.722 -bmeso

exp 0.320 80.000 145.000 67.300 -Relative error [%] 3.009 3.411 3.448 3.831 -

nFPtot 150, 000

0 0.2 0.4 0.60

0.5

1

1.5

2

2.5

δ

p D(δ)

(a)

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

×10−4

0

1

2

3

4

×104

` [m]

p L(`)

(b)Figure 8. Fixed-point iterative algorithm: probability density functions pD and pL of random variables Dand L, respectively. (a) pD(δ); (b) pL(`).

In terms of computational efficiency, we can see in Table 6 that the numbers of iterations n1, n2, n3

required to achieve the desired convergence are relatively low (less than or equal to 4) for each of the3 in silico test specimens yielding a number of calls to the deterministic numerical model at mesoscaleof 150, 000.

Finally, the results obtained for the identification of the parameters of the deterministic model atmacroscale and of the hyperparameters of the prior stochastic model at mesoscale for both validationexamples in 2D plane stress and 3D linear elasticity, for which the reference experimental values are knowna priori, demonstrate the efficiency, accuracy and robustness of the improved identification methodology,

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thereby allowing to apply it in the next section to a real biological material (beef femur cortical bone)with real experimental field measurements. Lastly, let us mention that the fixed-point iterative algorithmintroduced in the present work to solve the multi-objective optimization problem allows a considerablegain in terms of computational cost compared to the genetic algorithm used in Reference [140].

11. Numerical Application of the Multiscale Identification Method to Real Beef Cortical Bone inPlane Stress Linear Elasticity

In this section, we present a numerical application of the proposed multiscale identificationmethodology within the framework of 3D linear elasticity with 2D plane stress assumption by using a realexperimental database made up of 2D multiscale optical measurements of displacement fields (obtainedby DIC method) for only a single test specimen of cortical bone coming from a beef femur. The multiscaleexperimental test configuration corresponds to the one described in Section 3 and already considered inthe 2D and 3D numerical validation examples presented in Section 10. Technical details concerning theexperimental protocol (specimen preparation, measuring bench, optical image acquisition system and DICmethod) for obtaining the multiscale field measurements (performed simultaneously at both macroscaleand mesoscale) can be found in Reference [145]. The unique test specimen at macroscale is a cubic shapedsample with dimensions 1×1×1 cm3 prepared from bovine cortical bone. Even though such a biologicaltissue is often considered and modeled as a deterministic homogeneous medium with a transverselyisotropic linear elastic behavior at macroscale (>10 mm), its microstructure at mesoscale (from 500 µm to5 mm) contains randomly arranged osteons with some resorption cavities (lacuna), that are the principaltypes of inclusions/inhomogeneities, embedded in a matrix constituted by circumferential interstitiallamella surrounding Haversian canals. As a consequence, it is an anisotropic (heterogeneous) compositematerial with a complex hierarchical structure, which can be considered and modeled as a randomlinear elastic medium at mesoscale, and is therefore well adapted to the experimental application of themultiscale identification methodology developed in the present work. The single specimen is clamped onits lower face and loaded under vertical uniaxial compression onto its upper face with a maximal resultantforce of 9 kN so as to preserve a linear elastic material behavior. In order to reduce the measurementnoises (induced by the speckled pattern technique, the lighting of the observed 2D face, the optical imageacquisition system, etc.), a Gaussian spatial filter classically used in image processing has been appliedto smooth the experimental displacement fields umacro

exp = (umacroexp,1 , umacro

exp,2 ) and umesoexp = (umeso

exp,1, umesoexp,2)

measured at macroscale and at mesoscale, respectively. The images of experimental displacement fieldsat macroscale and at mesoscale have been filtered with a 2D Gaussian smoothing kernel with standarddeviation 3.5. This value has been chosen as a qualitative compromise allowing to regularize/smooththe experimental kinematic fields without removing the spatial fluctuations that are of the same order ofmagnitude as the lower bound of the search interval for the spatial correlation length `. Such a spatialfilter is also necessary to prevent the optimization algorithms from converging to a zero spatial correlationlength. Figures 9 and 10 represent the two components of macroscopic experimental displacement fieldumacro

exp over the 2D macroscopic domain Ωmacroexp and the ones of mesoscopic experimental displacement

field umesoexp over the 2D mesoscopic domain Ωmeso

exp , respectively, before and after application of the Gaussianspatial filter.

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(a) (b) (c) (d)Figure 9. Components umacro

exp,1 and umacroexp,2 of macroscopic experimental displacement field umacro

exp overthe 2D macroscopic domain Ωmacro

exp before and after application of the Gaussian spatial filter. (a) umacroexp,1

unfiltered; (b) umacroexp,1 filtered; (c) umacro

exp,2 unfiltered; (d) umacroexp,2 filtered.

(a) (b) (c) (d)Figure 10. Components umeso

exp,1 and umesoexp,2 of macroscopic experimental displacement field umeso

exp over the 2Dmesoscopic domain Ωmeso

exp before and after application of the Gaussian spatial filter. (a) umesoexp,1 unfiltered;

(b) umesoexp,1 filtered; (c) umeso

exp,2 unfiltered; (d) umesoexp,2 filtered.

11.1. Parameterization of the Macroscopic and Mesoscopic Models

In accordance with the experimental configuration and associated multiscale measurements, 2Dplane stresses are assumed and consequently, the deterministic and stochastic computational modelsat macroscale and mesoscale are the same as those used for the 2D validation example presented inSection 10.1. Thus, the modeling at macroscale and at mesoscale for the prior stochastic model, thehyperparameters and the parameterization are also exactly the same as in Section 10.1, namely definingSmeso in SFE+ and introducing vector-valued parameter a = (κ, µ) and vector-valued hyperparameterb = (δ, `, κ, µ). Optimal values of the latter are assumed to be in the reduced admissible set Bmeso

ad ⊂ Bmeso

constructed from information available in the literature such that δ ∈ [0.30 , 0.65], ` ∈ [50 , 100] µm,κ ∈ [9.5 , 11] GPa and µ ∈ [3.5 , 5.0] GPa, instead of the full admissible space Bmeso = ]0 , δsup[×]0 ,+∞[×]0 ,+∞[2 with δsup =

√(n + 1)/(n + 5) =

√7/11 ≈ 0.7977 < 1 (with n = 6 in linear elasticity). As in the

2D validation example, this reduced admissible set Bmesoad is discretized into nV = 10 points evenly spaced

in each dimension for which the three numerical indicators J mesoδ (b), J meso

` (b) and J multih (amacro, b)

defined in Section 6.4 are evaluated and compared.As for Section 10.1, both the deterministic boundary value problem (2)-(6) set on the macroscopic

domain Ωmacro and the stochastic boundary value problem (7)-(10) set on the mesoscopic domain Ωmeso

with 2D plane stress assumption are solved by discretizing the 2D domains of observation Ωmacroobs and

Ωmesoobs in space using the FEM. As 2D macroscopic and mesoscopic domains of observation Ωmacro

obs andΩmeso

obs are square domains, we consider for both a spatial discretization with a structured mesh madeup with 4-nodes linear quadrangular elements with Gauss-Legendre quadrature rule, in order to beconsistent with the regular grids used for the acquisition and discretization of experimental data. The2D macroscopic domain Ωmacro

obs with macroscale dimensions 1×1 cm2 is discretized with a structuredmesh of 9×9 = 81 quadrangular elements with uniform element size hmacro = 1.111 mm = 1.111×10−3 m

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in each spatial direction. The 2D mesoscopic domain Ωmesoobs with mesoscale dimensions 1×1 mm2 is

discretized with a structured mesh of 99×99 = 9801 quadrangular elements with uniform element sizehmeso = 10.10 µm = 1.010×10−5 m in each spatial direction. As for the 2D validation example, the sizeBRVE of representative volume element ΩRVE is defined with respect to the spatial correlation length ` suchthat BRVE = 20×`. The stochastic boundary value problem (7)-(10) at mesoscale is solved using the MonteCarlo numerical method and statistical convergence analyses have been systematically performed to setthe number of independent realizations for the statistical estimation of the mathematical expectationsinvolved in the different numerical indicators to the value Ns = 500.

11.2. Numerical Results of the Multiscale Statistical Inverse Identification

11.2.1. Resolution of the Single-Objective Optimization Problem at Macroscale

In this paragraph, we present the results of the first single-objective optimization problem (21) atmacroscale which consists in minimizing the macroscopic numerical indicator J macro(a) constructed in themacroscopic domain of observation Ωmacro

obs for identifying the optimal value amacro of a at macroscale. Thesingle-objective optimization problem (21) at macroscale has been solved using the Nelder-Mead simplexalgorithm. Table 7 gives the identified optimal value amacro = (11.335, 4.781) in [GPa], corresponding to amacroscopic transverse bulk modulus κmacro = 11.335 GPa and a macroscopic transverse shear modulusµmacro = 4.781 GPa, or equivalently to a macroscopic transverse Young’s modulus Emacro = 12.575 GPaand a macroscopic transverse Poisson’s ratio νmacro = 0.3151, which are in coherence with the valuesalready published and available in the literature for this type of biological material.

Table 7. Identified optimal value amacro of parameter a = (κ, µ).

κ [GPa] µ [GPa]

amacro 11.335 4.781

11.2.2. Resolution of the Multi-Objective Optimization Problem at Mesoscale

In this paragraph, we present the results of the second multi-objective optimization problem (22)at mesoscale which consists in simultaneously minimizing the three numerical indicators J meso

δ (b),J meso` (b) and J multi

h (amacro, b) constructed in the mesoscopic domain of observation Ωmesoobs using the

optimal parameter amacro = (11.335, 4.781) in [GPa] previously identified at macroscale (see the lastparagraph) for identifying the global optimal value bopt of b at mesoscale. The multi-objective optimizationproblem (22) has been solved only by using the fixed-point iterative algorithm (with a convergencecriterion on the residual norm between two iterations that must be less than a tolerance set to 10−9) andby searching for the solution of the multi-objective optimization problem (22) in a multidimensional gridof nV×nV×nV×nV points in the reduced admissible set Bmeso

ad ⊂ R4. The number of evaluations of thestochastic computational model needed by the fixed-point iterative algorithm is given by nFP

tot = 3 nV Ns nFP,where nFP is the number of iterations required to reach the desired convergence for the consideredmesoscopic domain of observation Ωmeso

obs .Table 8 gives the identified optimal value bmeso = (0.533, 61.111, 10.500, 4.667) in

([−], [µm], [GPa], [GPa]) obtained with the fixed-point iterative algorithm, corresponding to a dispersionparameter δmeso = 0.533, a spatial correlation length `meso = 61.111 µm, a mesoscopic mean transversebulk modulus κmeso = 10.500 GPa and a mesoscopic mean transverse shear modulus µmeso = 4.667 GPa,or equivalently to a mesoscopic mean transverse Young’s modulus Emeso = 12.194 GPa and a mesoscopicmean transverse Poisson’s ratio νmeso = 0.3064. The number of iterations nFP required to achieve the

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desired convergence with the fixed-point iterative algorithm over the mesoscopic subdomain Ωmeso isnFP = 5, leading to a number of evaluations of the stochastic computational model equal to nFP

tot = 7500.The identification results obtained at mesoscale are also in agreement with the information providedin the literature for this type of biological material. Indeed, from a physical standpoint, the identifiedspatial correlation length `meso = 61.111 µm turns out to be of the same order of magnitude as the distancebetween two adjacent lamellae of an osteon in bovine (beef femur) cortical bone. Moreover, such a valueof spatial correlation length is in accordance with the assumption of scale separation between macroscaleand mesoscale.

Table 8. Fixed-point iterative algorithm: identified optimal value bmeso of hyperparameter b = (δ, `, κ, µ)

for the mesoscopic domain of observation Ωmesoobs .

δ ` [µm] κ [GPa] µ [GPa] nFP

bmeso 0.533 61.111 10.500 4.667 5

nFPtot 7500

12. Conclusions

In the present work, we have revisited the multiscale identification methodology recently proposedin Reference [140] for the mechanical characterization of the apparent elastic properties of a complexmicrostructure made up of a heterogeneous anisotropic material that can be considered as a randomlinear elastic medium within the framework of 3D linear elasticity theory. Such a multiscale identificationhas been performed by solving a challenging multiscale statistical inverse problem (requiring multiscaleexperimental field measurements) formulated as a multi-objective optimization problem. This lattercan be decomposed into a first single-objective optimization problem defined at macroscale and asecond multi-objective optimization problem defined at mesoscale, to be solved sequentially andinvolving cost functions (numerical indicators) sufficiently sensitive to the variation of the parametersand hyperparameters to be identified. These numerical indicators allow for quantifying and minimizingthe distance between some relevant quantities of interest resulting from the multiscale experimentalfield measurements at macroscale and mesoscale on the one hand, and their counterparts obtainedthrough forward numerical simulations of a deterministic computational model at macroscale and of astochastic computational model at mesoscale corresponding to the experimental configuration on theother hand. We consider an ad hoc prior stochastic model introduced in Reference [144] for the numericalmodeling and simulation of the random elasticity field, which is parameterized by a small number ofhyperparameters. We also employ a stochastic computational homogenization method for the transfer ofstatistical information from mesoscale to macroscale. The multiscale identification methodology leads tothe identification of the optimal values of (i) the parameters involved in the deterministic model of theeffective (deterministic and homogeneous) elasticity tensor at macroscale and (ii) the hyperparametersinvolved in the prior stochastic model of the apparent (random and heterogeneous) elasticity tensor fieldat mesoscale

In the present paper, we have proposed two main improvements of the multiscale statistical inverseidentification methodology of the prior stochastic model. First, we have introduced an additionalsingle-objective cost function (numerical indicator) at mesoscale dedicated to the identification of the spatialcorrelation length(s) involved in the prior stochastic model, allowing the newly formulated multi-objectiveoptimization to be solved with a better computational efficiency by using a (computationally cheap)fixed-point iterative algorithm instead the (costly) global optimization algorithm (genetic algorithm)used in Reference [140]. The identification results obtained with the fixed-point iterative algorithm arepromising and comparable to that obtained with the genetic algorithm in terms of accuracy. Second,

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an ad hoc probabilistic modeling of the hyperparameters involved in the prior stochastic model andidentified on different mesoscopic domains of observation has been proposed in order to improve both therobustness and the precision of the statistical inverse identification method of the prior stochastic model.Finally, the improved identification methodology has been first validated on in silico materials within theframework of 2D plane stress and 3D linear elasticity with numerically simulated multiscale experimentaldata, and then successfully applied to real heterogeneous biological material within the framework of2D plane stress linear elasticity with real multiscale experimental measurements of 2D displacementfields obtained from a static uniaxial compression test performed on a single specimen made of bovinecortical bone and monitored by 2D digital image correlation at both macroscale and mesoscale. In linewith this work, several perspectives could be addressed: (i) the multi-objective optimization problemcould be solved by using machine learning based on artificial neural networks with a numerical databasegenerated from the stochastic computational model to train an artificial neural network in an (offline)preliminary phase and to use the trained neural network to perform the statistical inverse identificationin a computationally cheap (online) computing phase for further reducing the computational cost; (ii)the proposed methodology could be applied to real multiscale experimental measurements of full 3Ddisplacement fields obtained for example by X-ray computed microtomography and digital volumecorrelation, and also to other types of random heterogeneous materials; (iii) the proposed methodologycould be improved by identifying a posterior stochastic model of the non-Gaussian random elasticity (orcompliance) field in high stochastic dimension at the mesoscale of an anisotropic heterogeneous linearelastic microstructure using the identified prior stochastic model.

Author Contributions: Conceptualization, C.D.; methodology, C.D.; software, T.Z., C.D. and F.P.; validation, F.P. andC.D.; formal analysis, T.Z., F.P. and C.D.; investigation, T.Z.; resources, C.D. and F.P.; data curation, F.P. and C.D.;writing—original draft preparation, F.P., C.D. and T.Z.; writing—review and editing, F.P. and C.D.; visualization, F.P.;supervision, C.D. and F.P.; project administration, C.D.; funding acquisition, C.D. and F.P. All authors have read andagreed to the published version of the manuscript.

Funding: This research received no external funding.

Acknowledgments: The authors gratefully acknowledge Christian Soize, Professor at Université Gustave Eiffel,Laboratoire MSME, for helpful discussions and valuable suggestions.

Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

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a.s. almost surelyRVE Representative Volume ElementCCD Charge-Coupled DeviceCMOS Complementary Metal-Oxide-SemiconductorDIC Digital Image CorrelationDVC Digital Volume CorrelationµCT micro-Computed TomographyMRI Magnetic Resonance ImagingOCT Optical Coherence TomographyLS Least SquaresMLE Maximum Likelihood EstimationMaxEnt Maximum EntropyKL Karhunen-LoèvePC Polynomial ChaosFP Fixed-PointGA Genetic AlgorithmFEM Finite Element Method

References

1. Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties; Springer: New York, NY,USA, 2002; Volume 16, doi:10.1007/978-1-4757-6355-3.

2. Jeulin, D. Microstructure modeling by random textures. J. Microsc. Spectrosc. Electron. 1987, 12, 133–140.3. Jeulin, D. Morphological modeling of images by sequential random functions. Signal Process. 1989, 16, 403–431,

doi:10.1016/0165-1684(89)90033-9.4. Jeulin, D. Random texture models for material structures. Stat. Comput. 2000, 10, 121–132,

doi:10.1023/A:1008942325749.5. Jeulin, D. Random Structure Models for Homogenization and Fracture Statistics. In Mechanics of Random and

Multiscale Microstructures; Jeulin, D., Ostoja-Starzewski, M., Eds.; Springer: Vienna, Austria, 2001; pp. 33–91,doi:10.1007/978-3-7091-2780-3_2.

6. Jeulin, D. Morphology and effective properties of multi-scale random sets: A review. Comptes Rendus Mec. 2012,340, 219–229, doi:10.1016/j.crme.2012.02.004.

7. Pan, B.; Qian, K.; Xie, H.; Asundi, A. Two-dimensional digital image correlation for in-plane displacement andstrain measurement: A review. Meas. Sci. Technol. 2009, 20, 062001.

8. Sutton, M.A.; Orteu, J.J.; Schreier, H. Image Correlation for Shape, Motion and Deformation Measurements: BasicConcepts, Theory and Applications; Springer: Boston, MA, USA, 2009. doi:10.1007/978-0-387-78747-3.

9. Hild, F.; Roux, S. Optical Methods for Solid Mechanics. A Full-Field Approach; Chapter Digital Image Correlation;Wiley-VCH: Weinheim, Germany, 2012; pp. 183–228.

10. Kahn-Jetter, Z.L.; Jha, N.K.; Bhatia, H. Optimal image correlation in experimental mechanics. Opt. Eng. 1994,33, 1099–1105, doi:10.1117/12.166931.

11. Vendroux, G.; Knauss, W.G. Submicron deformation field measurements: Part 1. Developing a digital scanningtunneling microscope. Exp. Mech. 1998, 38, 18–23, doi:10.1007/BF02321262.

12. Vendroux, G.; Knauss, W.G. Submicron deformation field measurements: Part 2. Improved digital imagecorrelation. Exp. Mech. 1998, 38, 86–92, doi:10.1007/BF02321649.

13. Hild, F.; Roux, S. Digital Image Correlation: from Displacement Measurement to Identification of ElasticProperties—A Review. Strain 2006, 42, 69–80, doi:10.1111/j.1475-1305.2006.00258.x.

14. Roux, S.; Hild, F. Digital Image Mechanical Identification (DIMI). Exp. Mech. 2008, 48, 495–508,doi:10.1007/s11340-007-9103-3.

15. Réthoré, J.; Tinnes, J.P.; Roux, S.; Buffière, J.Y.; Hild, F. Extended three-dimensional digital image correlation(X3D-DIC). Comptes Rendus Mec. 2008, 336, 643–649, doi:10.1016/j.crme.2008.06.006.

37 of 43

16. Bornert, M.; Valès, F.; Gharbi, H.; Nguyen Minh, D. Multiscale Full-Field Strain Measurements forMicromechanical Investigations of the Hydromechanical Behaviour of Clayey Rocks. Strain 2010, 46, 33–46,doi:10.1111/j.1475-1305.2008.00590.x.

17. Constantinescu, A. On the identification of elastic moduli from displacement-force boundary measurements.Inverse Probl. Eng. 1995, 1, 293–313, doi:10.1080/174159795088027587.

18. Baxter, S.C.; Graham, L.L. Characterization of Random Composites Using Moving-Window Technique.J. Eng. Mech. 2000, 126, 389–397, doi:10.1061/(ASCE)0733-9399(2000)126:4(389).

19. Geymonat, G.; Hild, F.; Pagano, S. Identification of elastic parameters by displacement field measurement.Comptes Rendus Mec. 2002, 330, 403–408, doi:10.1016/S1631-0721(02)01476-6.

20. Geymonat, G.; Pagano, S. Identification of Mechanical Properties by Displacement Field Measurement:A Variational Approach. Meccanica 2003, 38, 535–545, doi:10.1023/A:1024766911435.

21. Graham, L.; Gurley, K.; Masters, F. Non-Gaussian simulation of local material properties based on amoving-window technique. Probabilistic Eng. Mech. 2003, 18, 223–234, doi:10.1016/S0266-8920(03)00026-2.

22. Bonnet, M.; Constantinescu, A. Inverse problems in elasticity. Inverse Probl. 2005, 21, R1–R50.23. Avril, S.; Pierron, F. General framework for the identification of constitutive parameters from full-field

measurements in linear elasticity. Int. J. Solids Struct. 2007, 44, 4978–5002, doi:10.1016/j.ijsolstr.2006.12.018.24. Avril, S.; Bonnet, M.; Bretelle, A.S.; Grédiac, M.; Hild, F.; Ienny, P.; Latourte, F.; Lemosse, D.; Pagano, S.; Pagnacco,

E.; et al. Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements. Exp.Mech. 2008, 48, 381, doi:10.1007/s11340-008-9148-y.

25. Flannery, B.P.; Deckman, H.W.; Roberge, W.G.; D’amico, K.L. Three-Dimensional X-ray Microtomography.Science 1987, 237, 1439–1444, doi:10.1126/science.237.4821.1439.

26. Kak, A.C.; Slaney, M. Principles of Computerized Tomographic Imaging; IEEE Press: Piscataway, NJ, USA, 1988.27. Baruchel, J.; Buffiere, J.Y.; Maire, E. X-Ray Tomography in Material Science; Hermes Science Publications: Paris,

France, 2000.28. Stock, S.R. Recent advances in X-ray microtomography applied to materials. Int. Mater. Rev. 2008, 53, 129–181,

doi:10.1179/174328008X277803.29. Desrues, J.; Viggiani, G.; Besuelle, P. Advances in X-ray Tomography for Geomaterials; John Wiley & Sons: Hoboken,

NJ, USA, 2010; Volume 118, doi:10.1002/9780470612187.30. Maire, E.; Withers, P.J. Quantitative X-ray tomography. Int. Mater. Rev. 2014, 59, 1–43,

doi:10.1179/1743280413Y.0000000023.31. van den Elsen, P.A.; Pol, E.D.; Viergever, M.A. Medical image matching-a review with classification. IEEE Eng.

Med. Biol. Mag. 1993, 12, 26–39. doi:10.1109/51.195938.32. Maintz, J.; Viergever, M.A. A survey of medical image registration. Med. Image Anal. 1998, 2, 1–36,

doi:10.1016/S1361-8415(01)80026-8.33. Liang, Z.P.; Lauterbur, P.C. Principles of Magnetic Resonance Imaging: A Signal Processing Perspective; IEEE Press

series in biomedical engineering; SPIE Optical Engineering Press: Bellingham Wash, WA, USA, 2000.34. Hill, D.L.G.; Batchelor, P.G.; Holden, M.; Hawkes, D.J. Medical image registration. Phys. Med. Biol. 2001,

46, R1–R45, doi:10.1088/0031-9155/46/3/201.35. Beaurepaire, E.; Boccara, A.C.; Lebec, M.; Blanchot, L.; Saint-Jalmes, H. Full-field optical coherence microscopy.

Opt. Lett. 1998, 23, 244–246, doi:10.1364/OL.23.000244.36. Schmitt, J.M. Optical coherence tomography (OCT): a review. IEEE J. Sel. Top. Quantum Electron. 1999,

5, 1205–1215, doi:10.1109/2944.796348.37. Fercher, A.F. Optical coherence tomography – development, principles, applications. Z. Med. Phys. 2010,

20, 251–276, doi:10.1016/j.zemedi.2009.11.002.38. Gambichler, T.; Jaedicke, V.; Terras, S. Optical coherence tomography in dermatology: technical and clinical

aspects. Arch. Dermatol. Res. 2011, 303, 457–473, doi:10.1007/s00403-011-1152-x.39. Popescu, D.P.; Choo-Smith, L.P.; Flueraru, C.; Mao, Y.; Chang, S.; Disano, J.; Sherif, S.; Sowa, M.G. Optical

coherence tomography: fundamental principles, instrumental designs and biomedical applications. Biophys. Rev.2011, 3, 155, doi:10.1007/s12551-011-0054-7.

38 of 43

40. Bay, B.K.; Smith, T.S.; Fyhrie, D.P.; Saad, M. Digital volume correlation: Three-dimensional strain mapping usingX-ray tomography. Exp. Mech. 1999, 39, 217–226, doi:10.1007/BF02323555.

41. Verhulp, E.; Rietbergen, B.; Huiskes, R. A three-dimensional digital image correlation technique for strainmeasurements in microstructures. J. Biomech. 2004, 37, 1313–1320, doi:10.1016/j.jbiomech.2003.12.036.

42. Bay, B.K. Methods and applications of digital volume correlation. J. Strain Anal. Eng. Des. 2008, 43, 745–760,doi:10.1243/03093247JSA436.

43. Roux, S.; Hild, F.; Viot, P.; Bernard, D. Three-dimensional image correlation from X-ray computed tomographyof solid foam. Compos. Part Appl. Sci. Manuf. 2008, 39, 1253–1265, doi:10.1016/j.compositesa.2007.11.011.

44. Rannou, J.; Limodin, N.; Réthoré, J.; Gravouil, A.; Ludwig, W.; Baïetto-Dubourg, M.C.; Buffière, J.Y.; Combescure,A.; Hild, F.; Roux, S. Three dimensional experimental and numerical multiscale analysis of a fatigue crack.Comput. Methods Appl. Mech. Eng. 2010, 199, 1307–1325, doi:10.1016/j.cma.2009.09.013.

45. Madi, K.; Tozzi, G.; Zhang, Q.; Tong, J.; Cossey, A.; Au, A.; Hollis, D.; Hild, F. Computation of full-fielddisplacements in a scaffold implant using digital volume correlation and finite element analysis. Med. Eng. Phys.2013, 35, 1298–1312, doi:10.1016/j.medengphy.2013.02.001.

46. Roberts, B.C.; Perilli, E.; Reynolds, K.J. Application of the digital volume correlation technique for themeasurement of displacement and strain fields in bone: A literature review. J. Biomech. 2014, 47, 923–934,doi:10.1016/j.jbiomech.2014.01.001.

47. Fedele, R.; Ciani, A.; Fiori, F. X-ray Microtomography under Loading and 3D-Volume Digital Image Correlation :A Review. Fundam. Inform. 2014, 135, 171–197, doi:10.3233/FI-2014-1117.

48. Hild, F.; Bouterf, A.; Chamoin, L.; Leclerc, H.; Mathieu, F.; Neggers, J.; Pled, F.; Tomicevic, Z.; Roux, S. Toward4D Mechanical Correlation. Adv. Model. Simul. Eng. Sci. 2016, 3, 17, doi:10.1186/s40323-016-0070-z.

49. Bouterf, A.; Adrien, J.; Maire, E.; Brajer, X.; Hild, F.; Roux, S. Identification of the crushing behavior of brittle foam:From indentation to oedometric tests. J. Mech. Phys. Solids 2017, 98, 181–200, doi:10.1016/j.jmps.2016.09.011.

50. Buljac, A.; Jailin, C.; Mendoza, A.; Neggers, J.; Taillandier-Thomas, T.; Bouterf, A.; Smaniotto, B.; Hild, F.;Roux, S. Digital Volume Correlation: Review of Progress and Challenges. Exp. Mech. 2018, 58, 661–708,doi:10.1007/s11340-018-0390-7.

51. Desceliers, C.; Ghanem, R.; Soize, C. Maximum likelihood estimation of stochastic chaos representations fromexperimental data. Int. J. Numer. Methods Eng. 2006, 66, 978–1001, doi:10.1002/nme.1576.

52. Ghanem, R.G.; Doostan, A. On the construction and analysis of stochastic models: Characterizationand propagation of the errors associated with limited data. J. Comput. Phys. 2006, 217, 63–81,doi:10.1016/j.jcp.2006.01.037.

53. Desceliers, C.; Soize, C.; Ghanem, R. Identification of Chaos Representations of Elastic Properties of RandomMedia Using Experimental Vibration Tests. Comput. Mech. 2007, 39, 831–838, doi:10.1007/s00466-006-0072-7.

54. Marzouk, Y.M.; Najm, H.N.; Rahn, L.A. Stochastic spectral methods for efficient Bayesian solution of inverseproblems. J. Comput. Phys. 2007, 224, 560–586, doi:10.1016/j.jcp.2006.10.010.

55. Arnst, M.; Clouteau, D.; Bonnet, M. Inversion of probabilistic structural models using measured transferfunctions. Comput. Methods Appl. Mech. Eng. 2008, 197, 589–608, doi:10.1016/j.cma.2007.08.011.

56. Das, S.; Ghanem, R.; Spall, J.C. Asymptotic Sampling Distribution for Polynomial Chaos Representation of Data:A Maximum Entropy and Fisher information approach. In Proceedings of the 45th IEEE Conference on Decisionand Control, San Diego, CA, USA, 13–15 December 2006; pp. 4139–4144, doi:10.1109/CDC.2006.377613.

57. Das, S.; Ghanem, R.; Finette, S. Polynomial chaos representation of spatio-temporal random fields fromexperimental measurements. J. Comput. Phys. 2009, 228, 8726–8751, doi:10.1016/j.jcp.2009.08.025.

58. Desceliers, C.; Soize, C.; Grimal, Q.; Talmant, M.; Naili, S. Determination of the random anisotropic elasticitylayer using transient wave propagation in a fluid-solid multilayer: Model and experiments. J. Acoust. Soc. Am.2009, 125, 2027–2034, doi:10.1121/1.3087428.

59. Guilleminot, J.; Soize, C.; Kondo, D. Mesoscale probabilistic models for the elasticity tensor of fiberreinforced composites: Experimental identification and numerical aspects. Mech. Mater. 2009, 41, 1309–1322,doi:10.1016/j.mechmat.2009.08.004.

39 of 43

60. Ma, X.; Zabaras, N. An efficient Bayesian inference approach to inverse problems based on an adaptive sparsegrid collocation method. Inverse Probl. 2009, 25, 035013.

61. Marzouk, Y.M.; Najm, H.N. Dimensionality reduction and polynomial chaos acceleration of Bayesian inferencein inverse problems. J. Comput. Phys. 2009, 228, 1862–1902, doi:10.1016/j.jcp.2008.11.024.

62. Arnst, M.; Ghanem, R.; Soize, C. Identification of Bayesian posteriors for coefficients of chaos expansions.J. Comput. Phys. 2010, 229, 3134–3154, doi:10.1016/j.jcp.2009.12.033.

63. Das, S.; Spall, J.C.; Ghanem, R. Efficient Monte Carlo computation of Fisher information matrix using priorinformation. Comput. Stat. Data Anal. 2010, 54, 272–289, doi:10.1016/j.csda.2009.09.018.

64. Ta, Q.A.; Clouteau, D.; Cottereau, R. Modeling of random anisotropic elastic media and impact on wavepropagation. Eur. J. Comput. Mech. 2010, 19, 241–253, doi:10.3166/ejcm.19.241-253.

65. Soize, C. Identification of high-dimension polynomial chaos expansions with random coefficients fornon-Gaussian tensor-valued random fields using partial and limited experimental data. Comput. MethodsAppl. Mech. Eng. 2010, 199, 2150–2164, doi:10.1016/j.cma.2010.03.013.

66. Soize, C. A computational inverse method for identification of non-Gaussian random fields using theBayesian approach in very high dimension. Comput. Methods Appl. Mech. Eng. 2011, 200, 3083–3099,doi:10.1016/j.cma.2011.07.005.

67. Lawson, C.; Hanson, R. Solving Least Squares Problems; Society for Industrial and Applied Mathematics:Philadelphia, PA, USA, 1995; doi:10.1137/1.9781611971217.

68. Soize, C. Uncertainty Quantification: An Accelerated Course with Advanced Applications in ComputationalEngineering, 1st ed.; Interdisciplinary Applied Mathematics; Springer: Berlin, Germany, 2017; Volume 47,doi:10.1007/978-3-319-54339-0.

69. Serfling, R. Approximation Theorems of Mathematical Statistics; Wiley: New York, NY, USA, 1980,doi:10.1002/9780470316481.

70. Papoulis, A.; Pillai, S.U. Probability, Random Variables, and Stochastic Processes, 4th ed.; McGraw-Hill HigherEducation: New York, NY, USA, 2002.

71. Spall, J.C. Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control; John Wiley & Sons:Hoboken, NJ, USA, 2005; Volume 65, doi:10.1002/0471722138.

72. Jaynes, E.T. Information Theory and Statistical Mechanics. Phys. Rev. 1957, 106, 620–630,doi:10.1103/PhysRev.106.620.

73. Jaynes, E.T. Information Theory and Statistical Mechanics. II. Phys. Rev. 1957, 108, 171–190,doi:10.1103/PhysRev.108.171.

74. Sobezyk, K.; Trebicki, J. Maximum entropy principle in stochastic dynamics. Probab. Eng. Mech. 1990, 5, 102–110,doi:10.1016/0266-8920(90)90001-Z.

75. Kapur, J.N.; Kesavan, H.K. Entropy Optimization Principles and Their Applications. In Entropy and EnergyDissipation in Water Resources; Singh, V.P., Fiorentino, M., Eds.; Springer: Dordrecht, The Netherlands, 1992; pp.3–20, doi:10.1007/978-94-011-2430-0_1.

76. Jumarie, G. Maximum Entropy, Information Without Probability and Complex Fractals: Classical and QuantumApproach; Fundamental Theories of Physics; Springer: Dordrecht, The Netherlands, 2000; Volume 112,doi:10.1007/978-94-015-9496-7.

77. Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: Cambridge, UK, 2003.78. Cover, T.M.; Thomas, J.A. Elements of Information Theory; A Wiley-Interscience Publication; Wiley: New York, NY,

USA, 2006.79. Bowman, A.W.; Azzalini, A. Applied Smoothing Techniques for Data Analysis; Oxford University Press: Oxford, UK,

1997.80. Beck, J.L.; Katafygiotis, L.S. Updating Models and Their Uncertainties. I: Bayesian Statistical Framework.

J. Eng. Mech. 1998, 124, 455–461, doi:10.1061/(ASCE)0733-9399(1998)124:4(455).81. Bernardo, J.M.; Smith, A.F.M. Bayesian Theory. Meas. Sci. Technol. 2001, 12, 221,

doi:10.1088/0957-0233/12/2/702.

40 of 43

82. Congdon, P. Bayesian Statistical Modelling, 2nd ed.; Wiley Series in Probability and Statistics; John Wiley and Sons,Ltd.: Chichester, UK, 2007.

83. Carlin, B.P.; Louis, T.A. Bayesian Methods for Data Analysis, 3rd ed.; Texts in Statistical Science; CRC Press: BocaRaton, FL, USA, 2009.

84. Stuart, A.M. Inverse problems: A Bayesian perspective. Acta Numer. 2010, 19, 451–559,doi:10.1017/S0962492910000061.

85. Tan, M.T.; Tian, G.L.; Ng, K.W. Bayesian Missing Data Problems: EM, Data Augmentation and NoniterativeComputation; Formerly CIP; Chapman & Hall/CRC Press: Boca Raton, FL, USA, 2010.

86. Rappel, H.; Beex, L.A.A.; Hale, J.S.; Noels, L.; Bordas, S.P.A. A Tutorial on Bayesian Inference to Identify MaterialParameters in Solid Mechanics. Arch. Comput. Methods Eng. 2020, 27, 361–385, doi:10.1007/s11831-018-09311-x.

87. Collins, J.D.; Hart, G.C.; Haselman, T.K.; Kennedy, B. Statistical Identification of Structures. AIAA J. 1974,12, 185–190, doi:10.2514/3.49190.

88. Walter, E.; Pronzato, L. Identification of Parametric Models: From Experimental Data, 1st ed.; Communications andControl Engineering; Springer: London, UK, 1997.

89. Kaipio, J.; Somersalo, E. Statistical and Computational Inverse Problems, 1st ed.; Applied Mathematical Sciences;Springer: New York, NY, USA, 2005; Volume 160, doi:10.1007/b138659.

90. Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation; Society for Industrial and AppliedMathematics: Philadelphia, PA, USA, 2005, doi:10.1137/1.9780898717921.

91. Isakov, V. Inverse Problems for Partial Differential Equations; Springer: Cham, Switzerland, 2006; Volume 127,doi:10.1007/978-3-319-51658-5.

92. Calvetti, D.; Somersalo, E. Inverse problems: From regularization to Bayesian inference. WIREs Comput. Stat.2018, 10, e1427, doi:10.1002/wics.1427.

93. Aster, R.C.; Borchers, B.; Thurber, C.H. Parameter Estimation and Inverse Problems, 3rd ed.; Elsevier: Amsterdam,The Netherlands, 2019, doi:10.1016/B978-0-12-804651-7.00016-X.

94. Karhunen, K. Zur Spektraltheorie Stochastischer Prozesse; Series A. 1, Mathematica-physica; Annales AcademiaeScientiarum Fennicae; 1946; Volume 34.

95. Loève, M. Probability Theory I. In Graduate Texts in Mathematics; Springer: New York, NY, USA, 1977; Volume 45.96. Loève, M. Probability Theory II. In Graduate Texts in Mathematics; Springer: New York, NY, USA, 1978; Volume

46.97. Ghanem, R.G.; Spanos, P.D. Stochastic Finite Elements: A Spectral Approach; Springer: New York, NY, USA, 1991,

doi:10.1007/978-1-4612-3094-6.98. Ghanem, R. Stochastic Finite Elements with Multiple Random Non-Gaussian Properties. J. Eng. Mech. 1999,

125, 26–40, doi:10.1061/(ASCE)0733-9399(1999)125:1(26).99. Xiu, D.; Karniadakis, G. The Wiener—Askey Polynomial Chaos for Stochastic Differential Equations. SIAM J.

Sci. Comput. 2002, 24, 619–644, doi:10.1137/S1064827501387826.100. Soize, C.; Ghanem, R. Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary

Probability Measure. SIAM J. Sci. Comput. 2004, 26, 395–410, doi:10.1137/S1064827503424505.101. Wan, X.; Karniadakis, G.E. Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures.

SIAM J. Sci. Comput. 2006, 28, 901–928, doi:10.1137/050627630.102. Xiu, D.; Hesthaven, J. High-Order Collocation Methods for Differential Equations with Random Inputs. SIAM J.

Sci. Comput. 2005, 27, 1118–1139, doi:10.1137/040615201.103. Babuška, I.; Nobile, F.; Tempone, R. A Stochastic Collocation Method for Elliptic Partial Differential Equations

with Random Input Data. SIAM J. Numer. Anal. 2007, 45, 1005–1034, doi:10.1137/050645142.104. Nouy, A. Generalized spectral decomposition method for solving stochastic finite element equations: Invariant

subspace problem and dedicated algorithms. Comput. Methods Appl. Mech. Eng. 2008, 197, 4718–4736,doi:10.1016/j.cma.2008.06.012.

105. Le Maître, O.P.; Knio, O.M. Spectral Methods for Uncertainty Quantification with Applications to Computational FluidDynamics; Springer: Dordrecht, The Netherlands, 2010, doi:10.1007/978-90-481-3520-2.

41 of 43

106. Caflisch, R.E. Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 1998, 7, 1–49,doi:10.1017/S0962492900002804.

107. Schuëller, G.; Spanos, P.D. Monte Carlo Simulation; A.A. Balkema: Rotterdam, The Netherlands, 2001.108. Rubinstein, R.Y.; Kroese, D.P. Simulation and the Monte Carlo method; Wiley Series in Probability and Statistics;

John Wiley & Sons: Hoboken, NJ, USA, 2016; Volume 10, doi:10.1002/9780470230381.109. Sanchez-Palencia, E. Non-Homogeneous Media and Vibration Theory, 1st ed.; Lecture Notes in Physics; Springer:

Berlin/ Heidelberg, Germany, 1986; Volume 127, doi:10.1007/3-540-10000-8.110. Sanchez-Palencia, E.; Zaoui, A. Homogenization Techniques for Composite Media; Lecture Notes in Physics; Springer:

Berlin/Heidelberg, Germany, 1985; Volume 272, p. IX, doi:10.1007/3-540-17616-0.111. Francfort, G.A.; Murat, F. Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal.

1986, 94, 307–334. doi:10.1007/BF00280908.112. Suquet, P.M. Introduction. In Homogenization Techniques for Composite Media; Sanchez-Palencia, E., Zaoui, A.,

Eds.; Springer: Berlin/Heidelberg, Germany, 1987; pp. 193–198.113. Huet, C. Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids

1990, 38, 813–841, doi:10.1016/0022-5096(90)90041-2.114. Sab, K. On the homogenization and the simulation of random materials. Eur. J. Mech. Solids 1992, 11, 585–607.115. Nemat-Nasser, S.; Hori, M. Micromechanics: Overall Properties of Heterogeneous Materials; North-Holland Series

in Applied Mathematics and Mechanics; North-Holland: Amsterdam, The Netherlands, 1993; Volume 37, pp.3–687.

116. Jikov, V.V.; Kozlov, S.M.; Oleinik, O.A. Homogenization of Differential Operators and Integral Functionals, 1st ed.;Springer: Berlin/Heidelberg, Germany, 1994; doi:10.1007/978-3-642-84659-5.

117. Suquet, P. Continuum Micromechanics, 1st ed.; CISM International Centre for Mechanical Sciences; Springer:Vienna, Austria, 1997; Volume 377, doi:10.1007/978-3-7091-2662-2.

118. Andrews, K.T.; Wright, S. Stochastic homogenization of elliptic boundary-value problems with Lp-data. Asymptot.Anal. 1998, 17, 165–184.

119. Pradel, F.; Sab, K. Homogenization of discrete media. J. Phys. IV France 1998, 8, Pr8-317–Pr8-324,doi:10.1051/jp4:1998839.

120. Forest, S.; Barbe, F.; Cailletaud, G. Cosserat modelling of size effects in the mechanical behaviour of polycrystalsand multi-phase materials. Int. J. Solids Struct. 2000, 37, 7105–7126, doi:10.1016/S0020-7683(99)00330-3.

121. Bornert, M.; Bretheau, T.; Gilormini, P. Homogénéisation en Mécanique des Matériaux 1. Matériaux Aléatoires élastiqueset Milieux Périodiques; Hermès Science publications: Paris, France, 2001.

122. Zaoui, A. Continuum Micromechanics: Survey. J. Eng. Mech. 2002, 128, 808–816,doi:10.1061/(ASCE)0733-9399(2002)128:8(808).

123. Kanit, T.; Forest, S.; Galliet, I.; Mounoury, V.; Jeulin, D. Determination of the size of the representative volumeelement for random composites: Statistical and numerical approach. Int. J. Solids Struct. 2003, 40, 3647–3679,doi:10.1016/S0020-7683(03)00143-4.

124. Bourgeat, A.; Piatnitski, A. Approximations of effective coefficients in stochastic homogenization. Ann. L’InstitutHenri Poincare (B) Probab. Stat. 2004, 40, 153–165, doi:10.1016/j.anihpb.2003.07.003.

125. Sab, K.; Nedjar, B. Periodization of random media and representative volume element size for linear composites.Comptes Rendus Mec. 2005, 333, 187–195, doi:10.1016/j.crme.2004.10.003.

126. Ostoja-Starzewski, M. Material spatial randomness: From statistical to representative volume element.Probabilistic Eng. Mech. 2006, 21, 112–132. doi:10.1016/j.probengmech.2005.07.007.

127. Ostoja-Starzewski, M. Microstructural Randomness and Scaling in Mechanics of Materials, 1st ed.; Chapman andHall/CRC Taylor & Francis: New York, NY, USA, 2007, doi:10.1201/9781420010275.

128. Soize, C. Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructureand probabilistic analysis of representative volume element size. Probab. Eng. Mech. 2008, 23, 307–323,doi:10.1016/j.probengmech.2007.12.019.

129. Xu, X.F.; Chen, X. Stochastic homogenization of random elastic multi-phase composites and size quantificationof representative volume element. Mech. Mater. 2009, 41, 174–186, doi:10.1016/j.mechmat.2008.09.002.

42 of 43

130. Tootkaboni, M.; Graham-Brady, L. A multi-scale spectral stochastic method for homogenization of multi-phaseperiodic composites with random material properties. Int. J. Numer. Methods Eng. 2010, 83, 59–90,doi:10.1002/nme.2829.

131. Guilleminot, J.; Noshadravan, A.; Soize, C.; Ghanem, R. A probabilistic model for bounded elasticity tensorrandom fields with application to polycrystalline microstructures. Comput. Methods Appl. Mech. Eng. 2011,200, 1637–1648, doi:10.1016/j.cma.2011.01.016.

132. Teferra, K.; Graham-Brady, L. A random field-based method to estimate convergence of apparentproperties in computational homogenization. Comput. Methods Appl. Mech. Eng. 2018, 330, 253–270,doi:10.1016/j.cma.2017.10.027.

133. Cottereau, R.; Clouteau, D.; Dhia, H.B.; Zaccardi, C. A stochastic-deterministic coupling method for continuummechanics. Comput. Methods Appl. Mech. Eng. 2011, 200, 3280–3288, doi:10.1016/j.cma.2011.07.010.

134. Desceliers, C.; Soize, C.; Naili, S.; Haiat, G. Probabilistic model of the human cortical bone with mechanicalalterations in ultrasonic range; Uncertainties in Structural Dynamics. Mech. Syst. Signal Process. 2012, 32, 170–177,doi:10.1016/j.ymssp.2012.03.008.

135. Clouteau, D.; Cottereau, R.; Lombaert, G. Dynamics of structures coupled with elastic media—A review ofnumerical models and methods. J. Sound Vib. 2013, 332, 2415–2436, doi:10.1016/j.jsv.2012.10.011.

136. Soize, C. Stochastic Models of Uncertainties in Computational Mechanics; Lecture Notes in Mechanics; AmericanSociety of Civil Engineers (ASCE): Reston, VA, USA, 2012; Volume 2.

137. Perrin, G.; Soize, C.; Duhamel, D.; Funfschilling, C. Identification of Polynomial Chaos Representations in HighDimension from a Set of Realizations. SIAM J. Sci. Comput. 2012, 34, A2917–A2945, doi:10.1137/11084950X.

138. Perrin, G.; Soize, C.; Duhamel, D.; Funfschilling, C. Karhunen–Loève expansion revisited forvector-valued random fields: Scaling, errors and optimal basis. J. Comput. Phys. 2013, 242, 607–622,doi:10.1016/j.jcp.2013.02.036.

139. Nouy, A.; Soize, C. Random field representations for stochastic elliptic boundary value problems and statisticalinverse problems. Eur. J. Appl. Math. 2014, 25, 339–373. doi:10.1017/S0956792514000072.

140. Nguyen, M.T.; Desceliers, C.; Soize, C.; Allain, J.M.; Gharbi, H. Multiscale identification of the random elasticityfield at mesoscale of a heterogeneous microstructure using multiscale experimental observations. Int. J. MultiscaleComput. Eng. 2015, 13, 281–295, doi:10.1615/IntJMultCompEng.2015011435.

141. Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 379–423,doi:10.1002/j.1538-7305.1948.tb01338.x.

142. Shannon, C.E. A Mathematical Theory of Communication. SIGMOBILE Mob. Comput. Commun. Rev. 2001,5, 3–55, doi:10.1145/584091.584093.

143. Balian, R. Random matrices and information theory. Il Nuovo Cimento B (1965–1970) 1968, 57, 183–193,doi:10.1007/BF02710326.

144. Soize, C. Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differentialoperators. Comput. Methods Appl. Mech. Eng. 2006, 195, 26–64, doi:10.1016/j.cma.2004.12.014.

145. Nguyen, M.T.; Allain, J.M.; Gharbi, H.; Desceliers, C.; Soize, C. Experimental multiscale measurements for themechanical identification of a cortical bone by digital image correlation. J. Mech. Behav. Biomed. Mater. 2016,63, 125–133, doi:10.1016/j.jmbbm.2016.06.011.

146. Cunha, N.D.; Polak, E. Constrained minimization under vector-valued criteria in finite dimensional spaces. J.Math. Anal. Appl. 1967, 19, 103–124, doi:10.1016/0022-247X(67)90025-X.

147. Censor, Y. Pareto optimality in multiobjective problems. Appl. Math. Optim. 1977, 4, 41–59,doi:10.1007/BF01442131.

148. Yu, P.L. Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, 1st ed.; Mathematical Conceptsand Methods in Science and Engineering; Springer: Boston, MA, USA, 1985; doi:10.1007/978-1-4684-8395-6.

149. Dauer, J.P.; Stadler, W. A survey of vector optimization in infinite-dimensional spaces, part 2. J. Optim. TheoryAppl. 1986, 51, 205–241, doi:10.1007/BF00939823.

150. Goldberg, D.E. Genetic Algorithms in Search, Optimization and Machine Learning, 1st ed.; Addison-Wesley LongmanPublishing Co., Inc.: Boston, MA, USA, 1989.

43 of 43

151. Deb, K. Multi-Objective Optimization using Evolutionary Algorithms; John Wiley & Sons: Chichester, UK, 2001.152. Marler, R.; Arora, J. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim.

2004, 26, 369–395, doi:10.1007/s00158-003-0368-6.153. Konak, A.; Coit, D.W.; Smith, A.E. Multi-objective optimization using genetic algorithms: A tutorial.

Special Issue—Genetic Algorithms and Reliability. Reliab. Eng. Syst. Saf. 2006, 91, 992–1007,doi:10.1016/j.ress.2005.11.018.

154. Coello Coello, C.A. Evolutionary multi-objective optimization: a historical view of the field. IEEE Comput. Intell.Mag. 2006, 1, 28–36, doi:10.1109/MCI.2006.1597059.

155. Coello, C.A.C.; Lamont, G.B.; Van Veldhuizen, D.A. Evolutionary Algorithms for Solving Multi-Objective Problems;Springer: Boston, MA, USA, 2007; Volume 5, doi:10.1007/978-0-387-36797-2.

156. Deb, K. Multi-objective Optimization. In Search Methodologies: Introductory Tutorials in Optimization andDecision Support Techniques; Burke, E.K., Kendall, G., Eds.; Springer: Boston, MA, USA, 2014; pp. 403–449,doi:10.1007/978-1-4614-6940-7_15.

157. Hughes, T.J.R. The Finite Element Method : Linear Static and Dynamic Finite Element Analysis; Prentice Hall:Englewood Cliffs, NJ, USA, 1987.

158. Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z. The Finite Element Method: Its Basis and Fundamentals;Butterworth-Heinemann: Oxford, UK, 2005.

159. Kalos, M.H.; Whitlock, P.A. Monte Carlo Methods Vol. 1: Basics; Wiley-Interscience: New York, NY, USA, 1986.160. Fishman, G.S. Monte Carlo: Concepts, Algorithms, and Applications, 1st ed.; Springer Series in Operations Research

and Financial Engineering; Springer: New York, NY, USA, 1996; doi:10.1007/978-1-4757-2553-7.161. Nelder, J.A.; Mead, R. A Simplex Method for Function Minimization. Comput. J. 1965, 7, 308–313,

doi:10.1093/comjnl/7.4.308.162. Walters, F.H.; Parker, L.R.; Morgan, S.L.; Deming, S.N. Sequential Simplex Optimization; CRC Press: Boca Raton,

FL, USA, 1991.163. Lagarias, J.; Reeds, J.; Wright, M.; Wright, P. Convergence Properties of the Nelder–Mead Simplex Method in

Low Dimensions. SIAM J. Optim. 1998, 9, 112–147, doi:10.1137/S1052623496303470.164. McKinnon, K. Convergence of the Nelder–Mead Simplex Method to a Nonstationary Point. SIAM J. Optim. 1998,

9, 148–158, doi:10.1137/S1052623496303482.165. Kolda, T.; Lewis, R.; Torczon, V. Optimization by Direct Search: New Perspectives on Some Classical and Modern

Methods. SIAM Rev. 2003, 45, 385–482, doi:10.1137/S003614450242889.166. Zadeh, L. Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control. 1963, 8, 59–60,

doi:10.1109/TAC.1963.1105511.167. Deb, K.; Sindhya, K.; Hakanen, J. MultiObjective Optimization. In Decision Sciences: Theory and Practice,

1st ed.; Sengupta, R.N., Gupta, A., Dutta, J., Eds.; CRC Press: Boca Raton, FL, USA, 2017; Chapter 3,doi:10.1201/9781315183176.

168. Guilleminot, J.; Soize, C. On the Statistical Dependence for the Components of Random Elasticity TensorsExhibiting Material Symmetry Properties. J. Elast. 2013, 111, 109–130, doi:10.1007/s10659-012-9396-z.